Heavy traffic analysis for EDF queues with reneging
aa r X i v : . [ m a t h . P R ] A p r The Annals of Applied Probability (cid:13)
Institute of Mathematical Statistics, 2011
HEAVY TRAFFIC ANALYSIS FOR EDF QUEUES WITHRENEGING
By Lukasz Kruk , John Lehoczky , Kavita Ramanan and Steven Shreve Maria Curie-Sklodowska University and Polish Academy of Sciences,Carnegie Mellon University, Brown University and Carnegie MellonUniversity
This paper presents a heavy-traffic analysis of the behavior of asingle-server queue under an Earliest-Deadline-First (EDF) schedul-ing policy in which customers have deadlines and are served onlyuntil their deadlines elapse. The performance of the system is mea-sured by the fraction of reneged work (the residual work lost dueto elapsed deadlines) which is shown to be minimized by the EDFpolicy. The evolution of the lead time distribution of customers inqueue is described by a measure-valued process. The heavy trafficlimit of this (properly scaled) process is shown to be a deterministicfunction of the limit of the scaled workload process which, in turn,is identified to be a doubly reflected Brownian motion. This papercomplements previous work by Doytchinov, Lehoczky and Shreve onthe EDF discipline in which customers are served to completion evenafter their deadlines elapse. The fraction of reneged work in a heav-ily loaded system and the fraction of late work in the correspondingsystem without reneging are compared using explicit formulas basedon the heavy traffic approximations. The formulas are validated bysimulation results.
1. Introduction.
Received December 2007; revised August 2009. Supported in part by the State Committee for Scientific Research of Poland, Grant 2P03A 012 23 and the EC FP6 Marie Curie ToK programme SPADE 2 at IMPAN, Poland. Supported in part by ONR and DARPA under MURI Contract N00014-01-1-0576. Supported in part by the NSF Grants DMS-04-06191 and DMS-04-05343 and CMMI-1059967 (formerly CMMI-0728064). Supported in part by the NSF Grants DMS-04-04682 and DMS-09-03475.
AMS 2000 subject classifications.
Primary 60K25; secondary 60G57, 60J65, 68M20.
Key words and phrases.
Due dates, heavy traffic, queueing, reneging, diffusion limits,random measures, real-time queues.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Applied Probability ,2011, Vol. 21, No. 2, 484–545. This reprint differs from the original in paginationand typographic detail. 1
KRUK, LEHOCZKY, RAMANAN AND SHREVE
Background and the reneging EDF model.
In the last decade, at-tention has been paid to queueing systems in which customers have dead-lines. Examples include telecommunication systems carrying digitized voiceor video traffic, tracking systems and real-time control systems. In the case ofvoice or video, packetized information must be received, processed and dis-played within stringent timing bounds so that the integrity of the transmis-sion is maintained. Similarly, there are processing requirements for trackingsystems that guarantee that a track can be successfully followed. Real-timecontrol systems (e.g., those associated with modern avionics systems, man-ufacturing plants or automobiles) also gather data that must be processedwithin stringent timing requirements in order for the system to maintain sta-bility or react to changes in the operating environment. We refer to queueingsystems that process tasks with deadlines as “real-time queueing systems.”The performance of a real-time queueing system is measured by its abil-ity to meet the deadlines of the customers. This is in contrast to ordinaryqueueing systems in which the measure of performance is often customerdelay, queue length or utilization of a service facility. We use the fractionof “reneged work,” defined as the residual work not serviced due to elapseddeadlines, as our performance measure. To minimize this quantity, it is nec-essary to use a scheduling policy that takes deadlines into account. We usethe Earliest-Deadline-First (EDF) policy, which reduces to the more familiarFirst-In-First-Out (FIFO) policy when all customers have the same dead-line. Under general assumptions, we prove that EDF is optimal with respectto this performance measure. A related result for
G/M/c queues, in whichthe number of reneging customers is used as a performance measure, wasobtained by Panwar and Towsley [29].Heavy traffic analysis of a single real-time queue was initiated by Lehocz-ky [27]. This was put on a firm mathematical foundation by Doytchinov,Lehoczky and Shreve (DLS) [7]. The accuracy of heavy traffic approxima-tions was developed in [22, 24]. The results of DLS were generalized to thecase of acyclic networks in [23]. In these papers it was assumed that allcustomers are served to completion. The case in which late customers leavethe system and their residual work is lost is addressed here. The main resultof this paper is a heavy traffic convergence theorem, from which is deriveda simple and practically useful approximation for the fraction of lost workwhen the system is heavily loaded.The mathematical formulation used by DLS and related papers is basedon random measures. In addition to the usual queue length and workloadprocesses associated with the queueing system, to model the evolution of areal-time queueing system, one must keep track of the lead time of each cus-tomer, that is, the time until the customer’s deadline elapses. This is done bymeasure-valued queue length and workload processes. The measure-valuedqueue length process puts unit mass on the real line at the lead time of each
DF QUEUES WITH RENEGING customer in the system, while the measure-valued workload process putsmass equal to the remaining service time of each customer at the lead timeof that customer. These measures evolve dynamically as customers arrive,age and depart. Under the usual heavy traffic assumptions, since customersare served to completion in the DLS framework, it is easy to see that theordinary scaled workload process converges weakly to a reflected Brown-ian motion with drift. DLS showed that the suitably scaled workload andqueue length measure-valued processes converge to an explicit deterministicfunction of the workload process.In this paper customers leave the system when their deadlines elapse,which we refer to as reneging. Due to the preemptive nature of the EDFpolicy, it is not possible to determine at the time of admission whether acustomer will be fully serviced before its deadline elapses. It is thus natu-ral to have the controller make the decision only at the time the deadlineelapses. The system with reneging shows marked improvement in perfor-mance over the DLS system, in the sense that the fraction of reneged workin this system is much less than the fraction of work that becomes late inthe DLS system. This improvement is because once a customer misses itsdeadline, the processor devotes no further effort to it, but rather turns itsattention to customers that are not late.The system with reneging is considerably more difficult to analyze thanthe DLS system. In the reneging system, the evolution of the scalar totalworkload process depends on the entire lead time distribution of customersin queue and the nature of the EDF discipline. This is in stark contrastto the DLS system, where the total workload process is independent of thescheduling discipline, and is identical to that of any GI/G/
KRUK, LEHOCZKY, RAMANAN AND SHREVE
Prediction formulas.
The results of this paper suggest a simple for-mula for the fraction of lost work in the EDF system with reneging. Inparticular, consider a single-server queue with traffic intensity ρ = λ/µ thatis near one, where 1 /λ is the mean interarrival time and 1 /µ is the meanservice time. Let α and β be the standard deviations of the interarrival timesand service times, respectively, and set σ = λ ( α + β ), which we assume isnonzero. Let D denote the mean lead time for arriving customers. Finally,set θ = 2(1 − ρ ) /σ . Under these circumstances,Fraction of lost work in reneging system ≈ e − θD (cid:18) − ρρ (1 − e − θD ) (cid:19) . (1.1)This formula is derived in Section 7.1 and compared with simulations inSection 7.2. If ρ = 1, in place of (1.1) we haveFraction of lost work in reneging system ≈ σ D . (1.2)Analysis of the limit of the standard (nonreneging) system suggests thatwhen ρ < ≈ e − θD , (1.3)which, together with (1.1), yields the approximationLost work in reneging systemLate work in standard system ≈ − ρρ (1 − e − θD ) . (1.4)If ρ ≥ ≈ σ D . (1.5)When plotted on a log scale, the fraction of lost work in the reneging systemand the fraction of late work in the standard system will be linear in D ,provided that e θD ≫
1, and these two plots will be separated by log((1 − ρ ) /ρ ). When performance is measured in terms of the work whose servicerequirement is not met by the time its deadline elapses, then the renegingsystem is far superior to the nonreneging system. We refer the reader to thesimulations in Section 7.2.The situation with reneged customers as opposed to reneged work is morecomplicated. DLS shows that the number of customers in the limiting stan-dard system at any time is just λ times the amount of work, the number oflate customers is λ times the amount of late work and henceFraction of late customers in the standard system(1.6) ≈ Fraction of late work in the standard system
DF QUEUES WITH RENEGING [see also (7.8) and its derivation for the case ρ < λ times the amount of ar-rived work and λ times the amount of work in the system (Corollary 3.7),respectively, but the number of customers who renege by a certain time isnot necessarily λ times the amount of reneged work by that time (see Re-mark 7.2). In particular, we do not have a formula like (1.6) for the renegingsystem. If the arrival process is Poisson, the fraction of lost customers in thereneging system can be estimated by a heuristic argument [see (7.7)] whichgives insteadFraction of lost customers in reneging system(1.7) ≈ µ β + 1 × (Fraction of lost work in reneging system) . Related work and outline of paper.
Measure-valued processes haverecently gained prominence in queueing theory. Decreusefond and Moyal[5] use such processes to obtain the fluid limit of an EDF
M/M/ √ n , they scale leadtimes by n and obtain a characterization of the limiting lead-time measure-valued process via a transport equation. In a different setting, Ward andGlynn [33, 34] find limits of FIFO queues with reneging. Measure-valuedprocesses have also proved useful in the heavy traffic analysis of queueswith scheduling disciplines other than EDF such as last-in-first-out [28],processor sharing [11, 12], and shortest remaining processing time [6, 13].As dynamical systems, queueing systems present a mathematical challengedue to discontinuities in their evolution at boundaries (which denote emptyqueues). The heavy traffic analysis of queueing systems described by R n -valued processes has been facilitated by the use of representations in termsof continuous mappings on R n [4, 8, 14, 31, 36]. This work demonstrates thatthis perspective can also be useful when the queueing system is representedby a more complicated, measure-valued process (see also [18] for recent workthat takes a similar perspective).Section 2 introduces our model. Section 3 summarizes the main results,and proofs of these results are given in Section 6. Section 4 introduces the ref-erence workload process and its decomposition, and describes its evolution.This reference workload process is easier to analyze than the workload pro-cess with reneging but the two are shown to have the same asymptotic behav-ior. Comparisons between the reference workload process and the renegingworkload process are presented in Section 5. Section 7 presents simulationresults. A proof of optimality of EDF, that may be of independent interest,is in the Appendix. KRUK, LEHOCZKY, RAMANAN AND SHREVE
2. The model, assumptions and notation.
Notation.
Let R be the set of real numbers. For a, b ∈ R , a ∨ b is themaximum of a and b , a ∧ b is the minimum and a + is the maximum of a and0. Also, inf { ∅ } should be understood as + ∞ , while sup { ∅ } and max { ∅ } should be understood as −∞ . Moreover, if a < b , then the interval [ b, a ] isunderstood to be ∅ .Denote by M the set of all finite, nonnegative measures on B ( R ), the Borelsubsets of R . Under the weak topology, M is a Polish space. We denote themeasure in M that puts one unit of mass at the point x ∈ R , that is, theDirac measure at x , by δ x . When ν ∈ M and B is an interval ( a, b ] or asingleton { a } , we will simply write ν ( a, b ] and ν { a } instead of ν (( a, b ]) and ν ( { a } ).Let T > X , we use D X [0 , ∞ ) (resp., D X [0 , T ]) to denote the space of right-continuous functions with left-handlimits (RCLL functions) from [0 , ∞ ) (resp., [0 , T ]) to X , equipped with theSkorokhod J topology. See [9] for details. When dealing with D X [0 , ∞ ) or D X [0 , T ], we typically consider X = R or R d , with appropriate dimension d for vector-valued functions, or X = M , unless explicitly stated otherwise.When X = R or M , for t > x ∈ D X [0 , ∞ ), we write x ( t − ) for the left-hand limit lim s ↑ t x ( s ), and we define △ x ( t ) to be the jump in x at time t , thatis, △ x ( t ) ∆ = x ( t ) − x ( t − ). Finally, given D X [0 , ∞ )-valued random variables Z n , n ∈ N , defined, respectively, on the probability spaces (Ω n , F n , P n ), n ∈ N , and a D X [0 , ∞ )-valued random variable Z defined on a probability space(Ω , F , P ), we say Z ( n ) converges in distribution to Z and write Z n ⇒ Z , iffor every bounded continuous function f on D X [0 , ∞ ), lim n →∞ E n [ f ( Z n )] = E [ f ( Z )]. Here E n and E are expectations taken with respect to P n and P ,respectively.2.2. The model with reneging.
We have a sequence of single-station queue-ing systems, each serving one class of customers. The queueing systemsare indexed by superscript ( n ). The inter-arrival times for the customersare { u ( n ) j } ∞ j =1 , a sequence of strictly positive, independent, identically dis-tributed random variables with common mean λ ( n ) and standard deviation α ( n ) . The service times are { v ( n ) j } ∞ j =1 , another sequence of positive, indepen-dent, identically distributed random variables with common mean µ ( n ) andstandard deviation β ( n ) .If the initial condition of the n th queue were not zero, then we would needto specify an initial workload measure-valued process and frontier [theseterms are defined in (2.17) and (2.19) below] in such a way that these havelimits under the heavy traffic scaling. However, if the limit of the initial DF QUEUES WITH RENEGING scaled workload process were not of the form appearing in Theorem 3.2below, then the workload process would be expected to have a jump at timezero. To avoid these complications, we assume that each queue is empty attime zero.We define the customer arrival times S ( n )0 ∆ = 0 , S ( n ) k ∆ = k X i =1 u ( n ) i , k ≥ , (2.1)the customer arrival process A ( n ) ( t ) ∆ = max { k ; S ( n ) k ≤ t } , t ≥ , (2.2)and the work arrival process V ( n ) ( t ) ∆ = ⌊ t ⌋ X j =1 v ( n ) j , t ≥ . (2.3)The work that has arrived to the queue by time t is then V ( n ) ( A ( n ) ( t )).Each customer arrives with an initial lead time L ( n ) j , the time betweenthe arrival time and the deadline for completion of service for that customer.These initial lead times are independent and identically distributed with P { L ( n ) j ≤ √ ny } = G ( y ) , (2.4)where G is a right-continuous cumulative distribution function. We define y ∗ ∆ = inf { y ∈ R | G ( y ) > } , y ∗ ∆ = min { y ∈ R | G ( y ) = 1 } (2.5)and assume that 0 < y ∗ ≤ y ∗ < + ∞ . We assume that for every n , the se-quences { u ( n ) j } ∞ j =1 , { v ( n ) j } ∞ j =1 and { L ( n ) j } ∞ j =1 are mutually independent. SeeRemark 3.9 for a discussion of these assumptions.We assume that customers are served using the Earliest-Deadline-First(EDF) queue discipline, that is, the customer with the shortest lead timereceives service. Preemption occurs when a customer more urgent than thecustomer in service arrives (we assume preempt-resume). There is no set up,switch-over, or other type of overhead. If the j th customer is still presentin the system (either waiting for service or receiving it) when his deadlinepasses, that is, at the time S ( n ) j + L ( n ) j , he leaves the queue immediately.This may be interpreted as either reneging or the result of an action of anexternal controller.We define W ( n ) ( t ), the workload process at time t , as the remaining pro-cessing time of all the customers in the system at this time. We define R ( n ) W ( t )to be the amount of work that reneges in the time interval [0 , t ]. The queuelength process Q ( n ) ( t ) is the number of customers in the queue at time t .The queueing system described above will be referred to as the EDF systemwith reneging . KRUK, LEHOCZKY, RAMANAN AND SHREVE
The standard EDF model.
We also have a sequence, indexed by su-perscript ( n ), of standard EDF systems , with the same stochastic primitivesas the EDF systems with reneging. In each of these standard systems, theserver serves the customer with the shortest lead time, preemption occursas in the reneging system, but late customers (customers with negative leadtimes) stay in the system until served to completion. The performance pro-cesses associated with the standard system will be denoted by the samesymbols as their counterparts from the system with reneging, but with ad-ditional subscript S . For example, W ( n ) S ( t ) denotes the workload in the stan-dard system at time t . The arrival processes A ( n ) ( t ) and V ( n ) ( t ) are the samefor the both systems, so we will not attach the subscript S to them.The standard EDF system is easier to analyze than the EDF system withreneging in several ways. For instance, the workload W ( n ) S in the standardsystem coincides with the workload of a corresponding G/G/1 queue (withthe same primitives) under any nonidling scheduling policy. More precisely,in the standard system the netput process N ( n ) ( t ) ∆ = V ( n ) ( A ( n ) ( t )) − t (2.6)measures the amount of work in queue at time t provided that the server isnever idle up to time t , and the cumulative idleness process I ( n ) S ( t ) ∆ = − inf ≤ s ≤ t N ( n ) ( s )(2.7)gives the amount of time the server is idle. Adding these two processestogether, we obtain the workload process for the standard system W ( n ) S ( t ) = N ( n ) ( t ) + I ( n ) S ( t ) . (2.8)(All the above processes are RCLL.) In contrast, the evolution of the work-load W ( n ) in the reneging system is more complex and depends not onlyon the residual service times but also on the lead times of all customers inthe queue. Our analysis of the reneging system will be facilitated by resultsfrom [7] on the heavy traffic analysis of the standard EDF system.2.4. Heavy traffic assumptions.
We assume that the following limits ex-ist: lim n →∞ λ ( n ) = λ, lim n →∞ µ ( n ) = λ, (2.9) lim n →∞ α ( n ) = α, lim n →∞ β ( n ) = β, and, moreover, λ > α + β >
0. Define the traffic intensity ρ ( n ) ∆ = λ ( n ) µ ( n ) .We make the heavy traffic assumption lim n →∞ √ n (1 − ρ ( n ) ) = γ (2.10) DF QUEUES WITH RENEGING for some γ ∈ R . We also impose the Lindeberg condition on the inter-arrivaland service times: for every c > n →∞ E [( u ( n ) j − ( λ ( n ) ) − ) I {| u ( n ) j − ( λ ( n ) ) − | >c √ n } ](2.11) = lim n →∞ E [( v ( n ) j − ( µ ( n ) ) − ) I {| v ( n ) j − ( µ ( n ) ) − | >c √ n } ] = 0 . We introduce the heavy traffic scaling for the idleness process in the stan-dard system and the workload and queue length processes for both EDFsystems b I ( n ) S ( t ) = 1 √ n I ( n ) S ( nt ) , c W ( n ) S ( t ) = 1 √ n W ( n ) S ( nt ) , b Q ( n ) S ( t ) = 1 √ n Q ( n ) S ( nt ) , c W ( n ) ( t ) = 1 √ n W ( n ) ( nt ) , b Q ( n ) ( t ) = 1 √ n Q ( n ) ( nt )and the centered heavy traffic scaling for the arrival processes b S ( n ) ( t ) = 1 √ n ⌊ nt ⌋ X j =1 (cid:18) u ( n ) j − λ ( n ) (cid:19) , b V ( n ) ( t ) = 1 √ n ⌊ nt ⌋ X j =1 (cid:18) v ( n ) j − µ ( n ) (cid:19) , b A ( n ) ( t ) = 1 √ n [ A ( n ) ( nt ) − λ ( n ) nt ] . The scaled netput process (which is the same for both systems) is given by b N ( n ) ( t ) = 1 √ n [ V ( n ) ( A ( n ) ( nt )) − nt ] . (2.12)Note that, by (2.8), c W ( n ) S ( t ) = b N ( n ) ( t ) + b I ( n ) S ( t ).It follows from Theorem 3.1 in [30] and Theorem 7.3.2 in [36] that( b S ( n ) , b A ( n ) ) ⇒ ( S ∗ , A ∗ ) , (2.13)where A ∗ is a zero-drift Brownian motion with variance α λ per unit timeand S ∗ ( λt ) = − λ A ∗ ( t ) , t ≥ . (2.14)It is a standard result [16] that( b N ( n ) , b I ( n ) S , c W ( n ) S ) ⇒ ( N ∗ , I ∗ S , W ∗ S ) , (2.15) KRUK, LEHOCZKY, RAMANAN AND SHREVE where N ∗ is a Brownian motion with variance ( α + β ) λ per unit time anddrift − γ , I ∗ S ( t ) ∆ = − min ≤ s ≤ t N ∗ ( s ) , W ∗ S ( t ) = N ∗ ( t ) + I ∗ S ( t ) . (2.16)In other words, W ∗ S is a Brownian motion reflected at 0 with variance ( α + β ) λ per unit time and drift − γ and I ∗ S causes the reflection.2.5. Measure-valued processes and frontiers.
To study whether tasks orcustomers meet their timing requirements, one must keep track of customerlead times. The action of the EDF discipline requires knowledge of the cur-rent lead times of all customers in system. We represent this information viaa collection of measure-valued stochastic processes.
Customer arrival measure-valued process : A ( n ) ( t )( B ) ∆ = (cid:26) Number of arrivals by time t , whether or not still in thesystem at time t , having lead times at time t in B ∈ B ( R ) (cid:27) . Workload arrival measure-valued process : V ( n ) ( t )( B ) ∆ = (cid:26) Work arrived by time t , whether or not still in the sys-tem at time t , having lead times at time t in B ∈ B ( R ) (cid:27) . Queue length measure-valued process : Q ( n ) ( t )( B ) ∆ = (cid:26) Number of customers in the queue at time t having lead times at time t in B ∈ B ( R ) (cid:27) . Workload measure-valued process : W ( n ) ( t )( B ) ∆ = (cid:26) Work in the queue at time t associated with cus-tomers having lead times at time t in B ∈ B ( R ) (cid:27) . (2.17)The latter two processes describe the behavior of the EDF system withreneging. Their counterparts for the standard EDF system will be denoted by Q ( n ) S ( t ) and W ( n ) S ( t ), respectively. The following relationships easily follow: A ( n ) ( t ) = A ( n ) ( t )( R ) , V ( n ) ( A ( n ) ( t )) = V ( n ) ( t )( R ) ,W ( n ) ( t ) = W ( n ) ( t )(0 , ∞ ) , Q ( n ) ( t ) = Q ( n ) ( t )(0 , ∞ ) ,W ( n ) S ( t ) = W ( n ) S ( t )( R ) , Q ( n ) S ( t ) = Q ( n ) S ( t )( R ) . In addition, we can represent the reneged work as follows: R ( n ) W ( t ) = X
0. The processes C ( n ) , F ( n ) and F ( n ) S are RCLL.We introduce heavy traffic scalings. For the real-valued processes Z ( n ) = C ( n ) , F ( n ) , F ( n ) S , W ( n ) , Q ( n ) , R ( n ) W , we define b Z ( n ) ( t ) ∆ = √ n Z ( n ) ( nt ) and for themeasure-valued processes Z ( n ) = Q ( n ) , W ( n ) , Q ( n ) S , W ( n ) S , A ( n ) , V ( n ) , we define b Z ( n ) ( t )( B ) ∆ = √ n Z ( n ) ( nt )( √ nB ) for every Borel set B ⊂ R .
3. Main results.
Before stating our main results, we summarize the re-sults for the standard EDF system that were obtained in [7]—in particular,we recall Proposition 3.10 and Theorem 3.1 of [7] which characterize the lim-iting distributions of the workload measure and the queue length measurein the standard system. Let H ( y ) ∆ = Z ∞ y (1 − G ( η )) dη = Z y ∗ y (1 − G ( η )) dη, if y ≤ y ∗ ,0 , if y > y ∗ .(3.1)The function H maps ( −∞ , y ∗ ] onto [0 , ∞ ) and is strictly decreasing andLipschitz continuous with Lipschitz constant 1 on ( −∞ , y ∗ ]. Therefore, thereexists a continuous inverse function H − that maps [0 , ∞ ) onto ( −∞ , y ∗ ]. Proposition 3.1 (Proposition 3.10 [7]).
We have b F ( n ) S ⇒ F ∗ S as n → ∞ ,where the limiting scaled frontier process F ∗ S for the standard EDF systemis explicitly given by F ∗ S ( t ) ∆ = H − ( W ∗ S ( t )) , t ≥ , (3.2) KRUK, LEHOCZKY, RAMANAN AND SHREVE with W ∗ S equal to Brownian motion with variance ( α + β ) λ per unit timeand drift − γ , reflected at . Theorem 3.2 (Theorem 3.1 [7]).
Let W ∗ S and Q ∗ S be the measure-valuedprocesses defined, respectively, by W ∗ S ( t )( B ) ∆ = Z B ∩ [ F ∗ S ( t ) , ∞ ) (1 − G ( y )) dy, Q ∗ S ( t )( B ) ∆ = λ W ∗ S ( t )( B ) , (3.3) for all Borel sets B ⊆ R . Then c W ( n ) S ⇒ W ∗S and b Q ( n ) S ⇒ Q ∗S , as n → ∞ . Remark 3.3.
The proofs in [7] can be modified to show that the con-vergences in (3.3) are in fact joint, that is, ( c W ( n ) S , b Q ( n ) S ) ⇒ ( W ∗ S , Q ∗ S ).There is lateness in the standard EDF system if and only if the measure-valued workload process has positive mass on the negative half line. Theo-rem 3.2 shows that, in the heavy traffic limit, this occurs exactly when thelimiting scaled frontier process F ∗ S lies to the left of 0 or, equivalently (byProposition 3.1), when W ∗ S is greater than H (0) = E [ L ( n ) j / √ n ], the mean ofthe scaled lead-time distribution. In the reneging system, there is no late-ness, and the amount of work that reneges is precisely the amount requiredto prevent lateness. Thus it is natural to expect that the limiting workloadin the reneging system will be constrained to remain below H (0). Let W ∗ be a Brownian motion with variance ( α + β ) λ per unit time and drift − γ ,reflected at 0 and H (0). The first main result of this paper is that W ∗ is thelimiting workload in the reneging system. Theorem 3.4. As n → ∞ , c W ( n ) ⇒ W ∗ . The next two results of this paper are the following counterparts of Propo-sition 3.1 and Theorem 3.2 for the EDF system with reneging.
Proposition 3.5.
We have b F ( n ) ⇒ F ∗ as n → ∞ , where F ∗ ( t ) ∆ = H − ( W ∗ ( t )) , t ≥ . (3.4)In other words, the process F ∗ defined by (3.4) is the limiting scaledfrontier process for the EDF system with reneging. Theorem 3.6.
Let W ∗ and Q ∗ be the measure-valued processes definedby W ∗ ( t )( B ) ∆ = Z B ∩ [ F ∗ ( t ) , ∞ ) (1 − G ( y )) dy, Q ∗ ( t )( B ) ∆ = λ W ∗ ( t )( B ) , (3.5) for all Borel sets B ⊆ R . Then ( c W ( n ) , b Q ( n ) ) ⇒ ( W ∗ , Q ∗ ) as n → ∞ . DF QUEUES WITH RENEGING By Theorem 3.6, the total masses of W ( n ) and Q ( n ) must converge jointlyto the total masses of W ∗ and Q ( n ) , respectively. Substituting B = R in (3.5)and using (3.1) and (3.4), we see that W ∗ ( t )( R ) = H ( F ∗ ( t )) = W ∗ ( t ) andwe recover Theorem 3.4. In fact, we have a stronger result. Corollary 3.7. As n → ∞ , ( c W ( n ) , b Q ( n ) ) ⇒ ( W ∗ , λW ∗ ) . Theorem 3.6 also shows that the limiting instantaneous lead-time profilesof customers in the EDF system with reneging conditioned on the value of the ( limiting ) workload in the system are the same as in the case of the standardEDF system. However, the limiting real-valued workload process for theEDF system with reneging is W ∗ , the doubly reflected Brownian motionand the unconditional limiting lead-time profiles for these two systems differaccordingly.We also have a characterization of the limiting amount of reneged work.
Theorem 3.8. As n → ∞ , b R ( n ) W ⇒ R ∗ W , where R ∗ W is the local time at H (0) of the doubly reflected Brownian motion W ∗ . Although these results are intuitive in light of the behavior of the standardEDF system, the proofs are challenging. Moreover, counter to what onemight expect, the result for queue lengths analogous to Theorem 3.8 is false.Specifically, although Corollary 3.7 shows that b Q ( n ) converges to the doublyreflected Brownian motion Q ∗ ∆ = λW ∗ on [0 , λH (0)], the scaled sequence b R ( n ) Q , n ∈ N , of reneged customers does not converge to the local time λR ∗ W of Q ∗ at λH (0). This observation, which is elaborated upon in Section 7,emphasizes the need for a rigorous justification of intuitive statements.The proof of Theorem 3.4 is in Section 6.1.1, the proofs of Proposition 3.5and Theorem 3.6 are in Section 6.1.2, and Section 6.2 contains the proof ofTheorem 3.8. We also establish an optimality property for EDF, Theorem5.1. Remark 3.9.
The assumption made in (2.5) that the support of thelead time distribution is bounded above by y ∗ < ∞ is mainly technical. Itis expected that the analysis in [21] for the standard EDF system under aweaker second moment condition can be applied to the reneging system aswell. On the other hand, the lower bound y ∗ > KRUK, LEHOCZKY, RAMANAN AND SHREVE to the density of the lead-time distribution near 0. From a modeling pointof view, it is reasonable to impose a strictly positive lower bound y ∗ >
4. The reference system.
In this section we introduce an auxiliary ref-erence workload measure-valued process U ( n ) and the corresponding real-valued reference workload process U ( n ) . In the special case of constant ini-tial lead times (i.e., y ∗ = y ∗ ), in which EDF reduces to the well-known FIFOservice discipline, U ( n ) and U ( n ) coincide with W ( n ) and W ( n ) , respectively.In general, these processes do not coincide (see Example 4.6), but, as we willshow in Section 6.1, the difference between the diffusion-scaled versions of U ( n ) and W ( n ) is negligible under heavy-traffic conditions. The advantage ofworking with the reference system, rather than the reneging system, is that U ( n ) can be represented explicitly as a certain mapping Φ of the measure-valued workload process W ( n ) S in the standard system. As shown in Section6.1, continuity properties of the mapping Φ enable an easy characterizationof the limiting distributions of U ( n ) and U ( n ) in heavy traffic.We begin with Section 4.1, where we define the reference system andprovide a useful decomposition of the process U ( n ) . In Section 4.2 we providea detailed description of the evolution of U ( n ) .4.1. Definition and properties of the reference workload.
In Section 4.1.1,we introduce a deterministic mapping on the space of measure-valued func-tions that is used to define the reference workload. Then, in Section 4.1.2,we provide a decomposition of the reference workload process.4.1.1.
A mapping Φ of measure-valued processes. We define a sequenceof reference workload measure-valued processes for the EDF system withreneging by the formula U ( n ) ∆ = Φ( W ( n ) S ) , (4.1)where the mapping Φ : D M [0 , ∞ ) D M [0 , ∞ ) is defined byΦ( µ )( t )( −∞ , y ](4.2) ∆ = h µ ( t )( −∞ , y ] − sup s ∈ [0 ,t ] (cid:16) µ ( s )( −∞ , ∧ inf u ∈ [ s,t ] µ ( u )( R ) (cid:17)i + DF QUEUES WITH RENEGING for every µ ∈ D M [0 , ∞ ), t ≥ y ∈ R . (The claim that Φ does indeedmap D M [0 , ∞ ) into D M [0 , ∞ ) is justified in Lemma 4.1 below.) We alsodefine the (real-valued) reference workload process U ( n ) as the total mass of U ( n ) , that is, U ( n ) ( t ) ∆ = U ( n ) ( t )( R ) ∀ t ∈ [0 , ∞ ) . (4.3)The frontier F ( n ) S defined in Section 2.3 played a crucial role in the descrip-tion and analysis of the evolution of the standard system in [7]. In a similarfashion, it will be useful to define the reference frontier E ( n ) ( t ) ∆ = (cid:26) inf { y ∈ R |U ( n ) ( t )( −∞ , y ] > } , if U ( n ) ( t ) > ∞ , if U ( n ) ( t ) = 0.(4.4)By definition, E ( n ) ( t ) is the leftmost point of support of the random measure U ( n ) ( t ) [understood as ∞ if U ( n ) ( t ) ≡ E ( n ) has RCLL paths.From (4.1)–(4.3) we have U ( n ) ( t )( −∞ , y ] = [ W ( n ) S ( t )( −∞ , y ] − K ( n ) ( t )] + , (4.5) U ( n ) ( t ) = W ( n ) S ( t ) − K ( n ) ( t ) , (4.6)where K ( n ) ( t ) ∆ = max s ∈ [0 ,t ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o . (4.7)In (4.7) we may write maximum rather than supremum because the pro-cess W ( n ) S ( · )( −∞ ,
0] never jumps down. Note from (4.6) and (4.7) that0 ≤ K ( n ) ( t ) ≤ W ( n ) S ( t ) and so for all t ≥ ≤ U ( n ) ( t ) ≤ W S ( t ) . (4.8)According to (4.6), the reference workload process U ( n ) is the standard work-load process W ( n ) S with mass K ( n ) removed. Equation (4.5) shows that thismass is removed from the left-hand side of the support of W ( n ) S . Moreover,since U ( n ) ( t )( −∞ , y ] > y to the right of the frontier E ( n ) ( t ), it isclear from (4.1) and (4.2) that for t ∈ [0 , ∞ ), y ≥ y > E ( n ) ( t ), U ( n ) ( t )( y , y ] = U ( n ) ( t )( −∞ , y ] − U ( n ) ( t )( −∞ , y ] = W ( n ) S ( t )( y , y ] , (4.9)which shows that U ( n ) coincides with W ( n ) S strictly to the right of E ( n ) .In the following lemma, we establish some basic properties of Φ that show,in particular, that U ( n ) ( t ), t ≥
0, and U ( n ) ( t ), t ≥
0, are stochastic processeswith sample paths in D M [0 , ∞ ) and D R + [0 , ∞ ), respectively. Although Φis not continuous on D M [0 , ∞ ), the lemma shows that it satisfies a certaincontinuity property that will be sufficient for our purposes. KRUK, LEHOCZKY, RAMANAN AND SHREVE
Lemma 4.1.
For every t ∈ [0 , ∞ ) , Φ( µ )( t )( −∞ ,
0] = 0 . Moreover, Φ maps D M [0 , ∞ ) to D M [0 , ∞ ) . Furthermore, if a sequence µ n , n ∈ N , in D M [0 , ∞ ) converges to µ ∈ D M [0 , ∞ ) , where µ is continuous and for every t ∈ [0 , ∞ ) , µ ( t ) { } = 0 , then Φ( µ n ) converges to Φ( µ ) in D M [0 , ∞ ) . Proof.
The first statement follows from the simple observation that,due to the nonnegativity of µ and (4.2),0 ≤ Φ( µ )( t )( −∞ , ≤ [ µ ( t )( −∞ , − µ ( t )( −∞ , ∧ µ ( t )( R )] + = 0 . Also, since the right-hand side of (4.2) is nondecreasing and right-continuousin y , we know that Φ( µ )( t ) ∈ M for every t ≥
0. Now, observe that Φ( µ )( t ) =Ψ( µ ( t ) , Γ( µ )( t )), where Ψ : M × R
7→ M is the mapping Ψ( ν, x )( −∞ , y ] ∆ =( ν ( −∞ , y ] − x ) + for all y ∈ R and Γ : D M [0 , ∞ ) R is defined byΓ( µ )( t ) ∆ = sup s ∈ [0 ,t ] (cid:16) µ ( s )( −∞ , ∧ inf u ∈ [ s,t ] µ ( u )( R ) (cid:17) ∀ t ∈ [0 , ∞ ) . Using the fact that weak convergence of measures on R is equivalent toconvergence of the cumulative distribution functions at continuity pointsof the limit, one can verify that Ψ is continuous on M × R . To show thatΦ( µ ) ∈ D M [0 , ∞ ), it suffices to show that Γ( µ ) ∈ D [0 , ∞ ). For this, we fix t ∈ [0 , ∞ ) and writeΓ( µ )( t + ε ) − Γ( µ )( t )= sup s ∈ [0 ,t ] h µ ( s )( −∞ , ∧ inf u ∈ [ s,t ] µ ( u )( R ) ∧ inf u ∈ [ t,t + ε ] µ ( u )( R ) i ∨ Z ( µ, ε )( t ) − sup s ∈ [0 ,t ] h µ ( s )( −∞ , ∧ inf u ∈ [ s,t ] µ ( u )( R ) i , where we define Z ( µ, ε )( t ) ∆ = sup s ∈ [ t,t + ε ] h µ ( s )( −∞ , ∧ inf u ∈ [ s,t + ε ] µ ( u )( R ) i . Since µ ∈ D M [0 , ∞ ) implies µ ( u ) converges weakly to µ ( t ) as u ↓ t , we havelim u ↓ t µ ( u )( R ) = µ ( t )( R ) and µ ( t )( −∞ , ≥ lim sup s ↓ t µ ( s )( −∞ ,
0] by Port-manteau’s theorem. This, in turn, implies that lim ε → Z ( µ, ε )( t ) = µ ( t )( −∞ , t ≥
0. Combining the above properties, it is easy to deduce thatΓ( µ )( t + ε ) − Γ( µ )( t ) → ε ↓
0, and the right-continuity of Φ( µ ) follows.The existence of left limits for Γ( µ ), and hence for Φ( u ), can be establishedby an analogous but simpler argument.Now, suppose µ n converges to µ in D M [0 , ∞ ) and µ is continuous with µ ( t ) { } = 0 for every t ≥
0. Then µ n ( t ) converges weakly to µ ( t ) uniformlyfor t in compact sets (u.o.c.) (see [2]). Since 0 is a continuity point for µ ( t ), DF QUEUES WITH RENEGING this implies µ n ( t )( −∞ ,
0] and µ n ( t )( R ) converge u.o.c. to µ ( t )( −∞ ,
0] and µ ( t )( R ), respectively. This shows that Γ( µ n )( t ) converges u.o.c. to Γ( µ )( t ),which, when combined with the continuity of Ψ, shows that Φ( µ n )( t ) con-verges weakly u.o.c. to Φ( µ )( t ). In particular, this shows Φ( µ n ) converges toΦ( µ ) in D M [0 , ∞ ). (cid:3) As an immediate consequence of the lemma, the definitions of U ( n ) and E ( n ) , and the fact that U ( n ) ( t ) is a purely atomic measure, we have, for all t ≥ U ( n ) ( t )( −∞ ,
0] = 0 and E ( n ) ( t ) > . (4.10)4.1.2. A decomposition of the reference workload.
We establish a decom-position of K ( n ) into its increasing and decreasing parts. Define σ ( n )0 ∆ = 0 and W ( n ) S (0 − ) ∆ = 0. For k = 0 , , , . . . , define recursively τ ( n ) k ∆ = min n t ≥ σ ( n ) k | W ( n ) S ( σ ( n ) k − ) ∨ max s ∈ [ σ ( n ) k ,t ] W ( n ) S ( s )( −∞ , ≥ W ( n ) S ( t ) o ,σ ( n ) k +1 ∆ = min { t ≥ τ ( n ) k | W ( n ) S ( t ) > W ( n ) S ( t − ) } . (4.12)In addition, for t ∈ [0 , ∞ ), define K ( n )+ ( t ) ∆ = X k ∈ N h W ( n ) S ( σ ( n ) k − ) ∨ max s ∈ [ σ ( n ) k ,t ∧ τ ( n ) k ] W ( n ) S ( s )( −∞ , − W ( n ) S ( σ ( n ) k − ) i ,K ( n ) − ( t ) ∆ = − X k ∈ N [( W ( n ) S ( τ ( n ) k − ) − ( σ ( n ) k ∧ t − τ ( n ) k − )) + (4.14) − W ( n ) S ( τ ( n ) k − )] . Theorem 4.2.
We have K ( n ) = K ( n )+ − K ( n ) − , (4.15) where K ( n )+ and K ( n ) − are the positive and negative variations of K ( n ) . More-over, Z [0 , ∞ ) I { U ( n ) ( s ) > } dK ( n ) − ( s ) = 0 . (4.16) KRUK, LEHOCZKY, RAMANAN AND SHREVE
The theorem is easily deduced from Propositions 4.3 and 4.4 and Remark4.5 below. The rest of the section is devoted to establishing these results.Observe that the late work W ( n ) S ( s )( −∞ ,
0] is right-continuous in s , re-maining constant or moving down at rate one and jumping up. There-fore, the maximum on the right-hand side of (4.11) is obtained. Addition-ally, because of the right-continuity of W ( n ) S and W ( n ) S , the minimum inthis equation is also obtained. Finally, W ( n ) S ( s )( −∞ ,
0] can never exceed W ( n ) S ( s ) = W ( n ) S ( s )( R ), and W ( n ) S never jumps down, so we must in facthave W ( n ) S ( σ ( n ) k − ) ∨ max s ∈ [ σ ( n ) k ,τ ( n ) k ] W ( n ) S ( s )( −∞ ,
0] = W ( n ) S ( τ ( n ) k ) . (4.17)For k ≥ σ ( n ) k is the first arrival time after τ ( n ) k − . We thus have W ( n ) S ( t ) = ( W ( n ) S ( τ ( n ) k − ) − ( t − τ ( n ) k − )) + , τ ( n ) k − ≤ t < σ ( n ) k . (4.18)We further have 0 = σ ( n )0 = τ ( n )0 < σ ( n )1 < τ ( n )1 < σ ( n )2 < · · · . (4.19) Proposition 4.3.
For each k ≥ , we have K ( n ) ( t ) = W ( n ) S ( σ ( n ) k − ) ∨ max s ∈ [ σ ( n ) k ,t ] W ( n ) S ( s )( −∞ , for t ∈ [ σ ( n ) k , τ ( n ) k ] . In particular, K ( n ) is nondecreasing on the interval [ σ ( n ) k , τ ( n ) k ] . Proof.
We proceed by induction on k . For the base case k = 1, notethat the standard EDF system is empty before the time σ ( n )1 . Therefore, W ( n ) S ( σ ( n )1 − ) = 0, and to prove (4.20), we must show that K ( n ) ( t ) = max s ∈ [0 ,t ] W ( n ) S ( s )( −∞ , , σ ( n )1 ≤ t ≤ τ ( n )1 . (4.21)For t ∈ [ σ ( n )1 , τ ( n )1 ], let s ( n ) ( t ) be the largest number in [ σ ( n )1 , t ] satisfying W ( n ) S ( s ( n ) ( t ))( −∞ ,
0] = max s ∈ [0 ,t ] W ( n ) S ( s )( −∞ , . (4.22)For u ∈ [ s ( n ) ( t ) , t ], we have W ( n ) S ( s ( n ) ( t ))( −∞ ,
0] = max s ∈ [ σ ( n )1 ,u ] W ( n ) S ( s )( −∞ , , DF QUEUES WITH RENEGING which is less than or equal to W ( n ) S ( u ) by the definition of τ ( n )1 and equation(4.17). Therefore,max s ∈ [0 ,t ] W ( n ) S ( s )( −∞ ,
0] = W ( n ) S ( s ( n ) ( t ))( −∞ , ≤ inf u ∈ [ s ( n ) ( t ) ,t ] W ( n ) S ( u ) . Equation (4.21) follows from (4.7).We assume (4.20) holds for some k and prove it for k + 1. For t ∈ [ σ ( n ) k +1 , τ ( n ) k +1 ], K ( n ) ( t ) = max s ∈ [0 ,σ ( n ) k +1 ) n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o (4.23) ∨ max s ∈ [ σ ( n ) k +1 ,t ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o . Equation (4.20) with k replaced by k + 1 will follow once we show thatmax s ∈ [0 ,σ ( n ) k +1 ) n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o = W ( n ) S ( σ ( n ) k +1 − )(4.24)and max s ∈ [ σ ( n ) k +1 ,t ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o (4.25) = max s ∈ [ σ ( n ) k +1 ,t ] W ( n ) S ( s )( −∞ , . For (4.24), we observe that because W ( n ) S ( s )( −∞ ,
0] and inf s ≤ u ≤ t W ( n ) S ( u ),regarded as functions of s , cannot increase except by a jump, the maximumon the left-hand side of (4.24) is attained. Let s ( n ) k be the largest number in[0 , σ ( n ) k +1 ) attaining this maximum. We havemax s ∈ [0 ,σ ( n ) k +1 ) n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o = W ( n ) S ( s ( n ) k )( −∞ , ∧ inf u ∈ [ s ( n ) k ,t ] W ( n ) S ( u ) ≤ W ( n ) S ( u ) ∀ u ∈ [ s ( n ) k , σ ( n ) k +1 ) , and so max s ∈ [0 ,σ ( n ) k +1 ) n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o ≤ W ( n ) S ( σ ( n ) k +1 − ) . (4.26)On the other hand, by the inequalities τ ( n ) k < σ ( n ) k +1 ≤ t ≤ τ ( n ) k +1 , definition(4.7), the induction hypothesis, and equation (4.17), we havemax s ∈ [0 ,σ ( n ) k +1 ) n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o KRUK, LEHOCZKY, RAMANAN AND SHREVE ≥ max s ∈ [0 ,τ ( n ) k ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,τ ( n ) k ] W ( n ) S ( u ) ∧ inf u ∈ [ τ ( n ) k ,t ] W ( n ) S ( u ) o = K ( n ) ( τ ( n ) k ) ∧ inf u ∈ [ τ ( n ) k ,t ] W ( n ) S ( u )= (cid:16) W ( n ) S ( σ ( n ) k − ) ∨ max s ∈ [ σ ( n ) k ,τ ( n ) k ] W ( n ) S ( s )( −∞ , (cid:17) ∧ inf u ∈ [ τ ( n ) k ,t ] W ( n ) S ( u )= W ( n ) S ( τ ( n ) k ) ∧ inf u ∈ [ τ ( n ) k ,t ] W ( n ) S ( u ) = inf u ∈ [ τ ( n ) k ,t ] W ( n ) S ( u ) . Equation (4.18) implies W ( n ) S ( u ) ≥ W ( n ) S ( σ ( n ) k +1 − ) for τ ( n ) k ≤ u < σ ( n ) k +1 . For σ ( n ) k +1 ≤ u ≤ t < τ ( n ) k +1 , (4.11) implies that W ( n ) S ( σ ( n ) k +1 − ) ∨ max s ∈ [ σ ( n ) k +1 ,u ] W ( n ) S ( s )( −∞ , ≤ W ( n ) S ( u ) , and so again we have W ( n ) S ( u ) ≥ W ( n ) S ( σ ( n ) k +1 − ). Finally, if u = t = τ ( n ) k +1 , then(4.17) implies that W ( n ) S ( u ) ≥ W ( n ) S ( σ ( n ) k +1 − ). It follows thatinf u ∈ [ τ ( n ) k ,t ] W ( n ) S ( u ) ≥ W ( n ) S ( σ ( n ) k +1 − ) . This gives the reverse of the inequality (4.26), and thus (4.24) is proved.For (4.25), we let t ( n ) k attain the maximum in max s ∈ [ σ ( n ) k +1 ,t ] W ( n ) S ( s )( −∞ , u ∈ [ t ( n ) k , t ], we have from (4.11) and (4.17) that W ( n ) S ( t ( n ) k )( −∞ ,
0] = max s ∈ [ σ ( n ) k +1 ,u ] W ( n ) S ( s )( −∞ , ≤ W ( n ) S ( u ) , and hence W ( n ) S ( t ( n ) k )( −∞ , ≤ inf u ∈ [ t ( n ) k ,t ] W ( n ) S ( u ). It follows thatmax s ∈ [ σ ( n ) k +1 ,t ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o ≤ W ( n ) S ( t ( n ) k )( −∞ ,
0] = W ( n ) S ( t ( n ) k )( −∞ , ∧ inf u ∈ [ t ( n ) k ,t ] W ( n ) S ( u ) ≤ max s ∈ [ σ ( n ) k +1 ,t ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o , which establishes (4.25). (cid:3) Proposition 4.4.
For each k ≥ , we have K ( n ) ( t ) = ( W ( n ) S ( τ ( n ) k − ) − ( t − τ ( n ) k − )) + , τ ( n ) k − ≤ t < σ ( n ) k . (4.27) DF QUEUES WITH RENEGING In particular, K ( n ) is nonincreasing on [ τ ( n ) k − , σ ( n ) k ) . Proof.
For all t ≥
0, we have K ( n ) ( t ) ≤ W ( n ) S ( t ), and for τ ( n ) k − ≤ t < σ ( n ) k ,we further have from (4.18) that K ( n ) ( t ) ≤ W ( n ) S ( t ) = ( W ( n ) S ( τ ( n ) k − ) − ( t − τ ( n ) k − )) + . (4.28)On the other hand, Proposition 4.3 and (4.17) with k replaced by k − s ∈ [0 ,τ ( n ) k − ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,τ ( n ) k − ] W ( n ) S ( u ) o = K ( n ) ( τ ( n ) k − ) = W ( n ) S ( σ ( n ) k − − ) ∨ max s ∈ [ σ ( n ) k − ,τ ( n ) k − ] W ( n ) S ( s )( −∞ , W ( n ) S ( τ ( n ) k − ) . For t ∈ [ τ ( n ) k − , σ ( n ) k ), it follows from (4.18) and the above equality that K ( n ) ( t ) = max s ∈ [0 ,t ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,t ] W ( n ) S ( u ) o ≥ max s ∈ [0 ,τ ( n ) k − ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,τ ( n ) k − ] W ( n ) S ( u ) ∧ inf u ∈ [ τ ( n ) k − ,t ] W ( n ) S ( u ) o = max s ∈ [0 ,τ ( n ) k − ] n W ( n ) S ( s )( −∞ , ∧ inf u ∈ [ s,τ ( n ) k − ] W ( n ) S ( u ) o (4.29) ∧ ( W ( n ) S ( τ ( n ) k − ) − ( t − τ ( n ) k − )) + = W ( n ) S ( τ ( n ) k − ) ∧ ( W ( n ) S ( τ ( n ) k − ) − ( t − τ ( n ) k − )) + = ( W ( n ) S ( τ ( n ) k − ) − ( t − τ ( n ) k − )) + . Equation (4.27) follows from (4.28) and (4.29). (cid:3)
Remark 4.5.
In light of (4.6) and Proposition 4.3, we have the charac-terization of τ ( n ) k as τ ( n ) k = min { t ≥ σ ( n ) k | K ( n ) ( t ) ≥ W ( n ) S ( t ) } = min { t ≥ σ ( n ) k | U ( n ) ( t ) = 0 } . (4.30)Because σ ( n ) k +1 is the time of first arrival after τ ( n ) k , we in fact have U ( n ) ( t ) = 0 , τ ( n ) k ≤ t < σ ( n ) k +1 . (4.31) KRUK, LEHOCZKY, RAMANAN AND SHREVE
Evaluating (4.20) at σ ( n ) k and using W ( n ) S ( σ ( n ) k − ) ≥ W ( n ) S ( σ ( n ) k )( −∞ , K ( n ) ( σ ( n ) k ) = W ( n ) S ( σ ( n ) k − ) . (4.32)But (4.18) and Proposition 4.4 show that K ( n ) ( σ ( n ) k − ) = W ( n ) S ( σ ( n ) k − ) , (4.33)and so △ K ( n ) ( σ ( n ) k ) = 0 . (4.34)By contrast △ K ( n ) ( τ ( n ) k ) can be positive. Evaluating (4.20) at τ ( n ) k and using(4.17), we obtain K ( n ) ( τ ( n ) k ) = W ( n ) S ( τ ( n ) k ) . (4.35)In conclusion, K ( n ) ( t ) = K ( n ) ( σ ( n ) k ) ∨ max s ∈ [ σ ( n ) k ,t ] W ( n ) S ( s )( −∞ , , σ ( n ) k ≤ t ≤ τ ( n ) k , (4.36) K ( n ) ( t ) = ( K ( n ) ( τ ( n ) k − ) − ( t − τ ( n ) k − )) + , τ ( n ) k − ≤ t < σ ( n ) k . (4.37)4.2. Dynamics of the reference workload process.
The evolutions of U ( n ) and W ( n ) are similar; the difference between them is asymptotically negli-gible. Before proving the properties of U ( n ) , we provide a summary of theseproperties. The reader may work out the evolution of W ( n ) S , U ( n ) and W ( n ) in Example 4.6 to follow along. This example appears in detail in [26]. Example 4.6.
Consider a system realization in which u ( n )1 = 1 , v ( n )1 = 4 , L ( n )1 = 3 , S ( n )1 = 1 ,u ( n )2 = 1 , v ( n )2 = 4 , L ( n )2 = 5 , S ( n )2 = 2 ,u ( n )3 = 3 , v ( n )3 = 2 , L ( n )3 = 1 , S ( n )3 = 5 ,u ( n )4 = 2 , v ( n )4 = 1 , L ( n )4 = 4 , S ( n )4 = 7 ,u ( n )5 = 2 , v ( n )5 = 1 , L ( n )5 = 1 , S ( n )5 = 9 . Recall that δ a is a unit point mass at a . It is straightforward to compute W ( n ) S ( t ) = , ≤ t < − t ) δ − t , ≤ t < − t ) δ − t + 4 δ − t , ≤ t < − t ) δ − t + 4 δ − t , ≤ t < − t ) δ − t + δ − t , ≤ t < δ − + δ + δ , t = 9, DF QUEUES WITH RENEGING U ( n ) ( t ) = , ≤ t < − t ) δ − t , ≤ t < − t ) δ − t + 4 δ − t , ≤ t < − t ) δ − t , ≤ t < − t ) δ − t + 4 δ − t , ≤ t < − t ) δ − t , ≤ t < − t ) δ − t , ≤ t < , ≤ t < δ , t = 9, W ( n ) ( t ) = , ≤ t < − t ) δ − t , ≤ t < − t ) δ − t + 4 δ − t , ≤ t < − t ) δ − t , ≤ t < − t ) δ − t + 3 δ − t , ≤ t < − t ) δ − t , ≤ t < − t ) δ − t , ≤ t < , ≤ t < δ , t = 9.Recall that K ( n ) is the amount of mass removed from the standard work-load W ( n ) S to obtain the reference workload U ( n ) . To understand the process K ( n ) , we consider the dynamics of U ( n ) . In the absence of new arrivals, allatoms of U ( n ) move left with unit speed. Moreover, the mass of the leftmostatom of U ( n ) decreases with unit speed until it vanishes, corresponding tothe work being done on the most urgent job in queue until it is served tocompletion [Proposition 4.8(i)]. However, if the leftmost atom of U ( n ) hitszero, this atom is immediately removed from U ( n ) [Proposition 4.8(ii), (v)].This may be interpreted as reneging of a customer or deletion of a late cus-tomer from the system. When there is a new arrival at time t with lead timenot smaller than the leftmost point of support of U ( n ) ( t − ), and this point ofsupport is strictly positive, then a mass of the size v ( n ) A ( n ) ( t ) located at L ( n ) A ( n ) ( t ) is added to U ( n ) ( t − ) [Proposition 4.8(iii)]. Similarly, if there is a new arrivaland the leftmost point of the support of U ( n ) hits zero at the same time,then both of the above actions take place [(4.53) of Proposition 4.8(v)]. Thisis the case of a simultaneous new arrival and ejection of a late customer. TheEDF system with reneging W ( n ) shows the same behavior in all these cases.However, if a customer arrives to start a new busy period for U ( n ) or, if attime t , there is a new arrival with lead time more urgent than the leftmostpoint of the support of U ( n ) ( t − ) (i.e., we have a “preemption”), then the KRUK, LEHOCZKY, RAMANAN AND SHREVE mass v ( n ) A ( n ) ( t ) associated with the new arrival is distributed in [ L ( n ) A ( n ) ( t ) , ∞ ),or more precisely, on some atoms of W ( n ) S ( t ) located on this half-line, but itis not necessarily located at the single atom L ( n ) A ( n ) ( t ) . This possibility is de-scribed in Lemma 4.7 and Proposition 4.8(iv). In this respect, the evolutionof U ( n ) differs from that of W ( n ) , for which all the new mass is always placedat the lead time of the arriving customer. Example 4.6 illustrates this.We now begin the rigorous study of U ( n ) . As shown in Section 4.1, thetime interval [0 , ∞ ) can be decomposed into a union of the disjoint intervals( τ ( n ) k , σ ( n ) k +1 ] and ( σ ( n ) k , τ ( n ) k ], k ≥
0, such that K ( n ) = W ( n ) S − U ( n ) is non-increasing on ( τ ( n ) k , σ ( n ) k +1 ] and nondecreasing on ( σ ( n ) k , τ ( n ) k ]. In Lemma 4.7below, we analyze the behavior of U ( n ) on the time intervals [ τ ( n ) k − , σ ( n ) k ], k ≥
1, while Proposition 4.8 describes the dynamics of U ( n ) on the intervals( σ ( n ) k , τ ( n ) k ), k ≥
1. The section ends with Corollary 4.9, which describes thetime evolution of the reference workload process U ( n ) .We make use of the following elementary facts about the standard work-load. Since the interarrival times are strictly positive, △ A ( n ) ( t ) ∈ { , } , and W ( n ) S ( t ) = W ( n ) S ( t − ) + △ A ( n ) ( t ) v ( n ) A ( n ) ( t ) δ L ( n ) A ( n )( t ) , t ≥ , (4.38)which implies △ W ( n ) S ( t ) = △ A ( n ) ( t ) v ( n ) A ( n ) ( t ) , t ≥ . (4.39)For any functions f and g on [0 , ∞ ) (taking finite or infinite values) suchthat whenever s < t and t − s is small enough, f ( s ) = f ( t − ) + t − s and g ( s ) = g ( t − ) + t − s , we havelim s ↑ t W ( n ) S ( s )[ f ( s ) , g ( s )] = W ( n ) S ( t − )[ f ( t − ) , g ( t − )] . (4.40)This is true because the lead times of the customers present in the stan-dard system decrease with unit rate. Equation (4.40) remains valid if theclosed intervals [ f ( · ) , g ( · )] are replaced by either [ f ( · ) , g ( · )), ( f ( · ) , g ( · )] or( f ( · ) , g ( · )). These facts will be used repeatedly in the following arguments,sometimes without mention. Lemma 4.7.
Let k ≥ . We have U ( n ) ( t ) = 0 , τ ( n ) k − ≤ t < σ ( n ) k , (4.41) △ U ( n ) ( σ ( n ) k ) = v ( n ) A ( n ) ( σ ( n ) k ) , (4.42) U ( n ) ( σ ( n ) k )( −∞ , L ( n ) A ( n ) ( σ ( n ) k ) ) = 0 . (4.43) DF QUEUES WITH RENEGING Proof.
Equation (4.41) follows immediately from (4.6), (4.18) and Propo-sition 4.4. By (4.6), (4.34), (4.39) and the fact that △ A ( n ) ( σ ( n ) k ) = 1, we have △ U ( n ) ( σ ( n ) k ) = △ W ( n ) S ( σ ( n ) k ) − △ K ( n ) ( σ ( n ) k ) = v ( n ) A ( n ) ( σ ( n ) k ) , and (4.42) follows. For y < L ( n ) A ( n ) ( σ ( n ) k ) , (4.5), (4.38), (4.34) and (4.33) imply U ( n ) ( σ ( n ) k )( −∞ , y ] = [ W ( n ) S ( σ ( n ) k )( −∞ , y ] − K ( n ) ( σ ( n ) k )] + = [ W ( n ) S ( σ ( n ) k − )( −∞ , y ] − K ( n ) ( σ ( n ) k − )] + ≤ [ W ( n ) S ( σ ( n ) k − ) − K ( n ) ( σ ( n ) k − )] + = 0 , and so (4.43) also follows. (cid:3) Lemma 4.7 shows that σ ( n ) k begins a busy period for the reference system.Equation (4.30) implies that U ( n ) ( t ) > σ ( n ) k < t < τ ( n ) k , and thus theintervals [ σ ( n ) k , τ ( n ) k ), k ≥
1, are precisely the busy periods for the referencesystem. We analyze the behavior of U ( n ) during these busy periods. We startwith the observation that, by (4.5) and Proposition 4.3, for t ∈ ( σ ( n ) k , τ ( n ) k ), U ( n ) ( t )( −∞ , y ] = h W ( n ) S ( t )( −∞ , y ](4.44) − (cid:16) W ( n ) S ( σ ( n ) k − ) ∨ max s ∈ [ σ ( n ) k ,t ] W ( n ) S ( s )( −∞ , (cid:17)i + . In what follows, given ν ∈ M and any interval I ⊂ R , we will use ν |I to denote the measure in M that is zero on I c and coincides with ν on I : ν |I ( B ) = ν ( B ∩ I ) for all B ∈ B ( R ). Proposition 4.8.
For k ≥ and σ ( n ) k < t < τ ( n ) k , the following five prop-erties hold: (i) If △ A ( n ) ( t ) = 0 and E ( n ) ( t − ) > , then △ K ( n ) ( t ) = 0 , (4.45) △ U ( n ) ( t ) = 0 . (4.46) In this case, if U ( n ) ( t − ) { E ( n ) ( t − ) } > , then both U ( n ) ( · ) { E ( n ) ( · ) } and U ( n ) ( t ) decrease with unit rate in a neighborhood of t . On the other hand, if U ( n ) ( t − ) { E ( n ) ( t − ) } = 0 , then U ( n ) ( t ) = W ( n ) S ( t ) | [ E ( n ) ( t ) , ∞ ) . KRUK, LEHOCZKY, RAMANAN AND SHREVE (ii) If △ A ( n ) ( t ) = 0 and E ( n ) ( t − ) = 0 , then U ( n ) ( t − ) { } = △ K ( n ) ( t ) = − △ U ( n ) ( t )(4.47) and U ( n ) ( t ) = W ( n ) S ( t ) | (0 , ∞ ) . (iii) If △ A ( n ) ( t ) = 1 and L ( n ) A ( n ) ( t ) ≥ E ( n ) ( t − ) > , then (4.45) holds, △ E ( n ) ( t ) ≥ and U ( n ) ( t ) = U ( n ) ( t − ) + v ( n ) A ( n ) ( t ) δ L ( n ) A ( n )( t ) . (4.48)(iv) If △ A ( n ) ( t ) = 1 , L ( n ) A ( n ) ( t ) < E ( n ) ( t − ) , then (4.45) holds and L ( n ) A ( n ) ( t ) ≤ E ( n ) ( t ) ≤ E ( n ) ( t − ) , (4.49) △ U ( n ) ( t ) = v ( n ) A ( n ) ( t ) , (4.50) U ( n ) ( t ) | ( E ( n ) ( t − ) , ∞ ) = U ( n ) ( t − ) | ( E ( n ) ( t − ) , ∞ ) , (4.51) U ( n ) ( t ) { E ( n ) ( t − ) } ≥ U ( n ) ( t − ) { E ( n ) ( t − ) } . (4.52)(v) If △ A ( n ) ( t ) = 1 and L ( n ) A ( n ) ( t ) > E ( n ) ( t − ) = 0 , then U ( n ) ( t ) = U ( n ) ( t − ) + v ( n ) A ( n ) ( t ) δ L ( n ) A ( n )( t ) − U ( n ) ( t − ) { } δ . (4.53) Proof.
Fix k ≥ t ∈ ( σ ( n ) k , τ ( n ) k ). We start with the general obser-vation that, by (4.4) and (4.44), E ( n ) ( t ) = min n y |W ( n ) S ( t )( −∞ , y ](4.54) > W ( n ) S ( σ ( n ) k − ) ∨ max s ∈ [ σ ( n ) k ,t ] W ( n ) S ( s )( −∞ , o , and because W ( n ) S ( t ) is purely atomic, the minimum on the right-hand sideof (4.54) is obtained at the atom of W ( n ) S ( t ) located at y = E ( n ) ( t ). Inparticular, W ( n ) S ( t ) { E ( n ) ( t ) } > . (4.55)We now consider each of the five different cases of the proposition.(i) Let a = E ( n ) ( t − ). By (4.4) and (4.5), for all s < t sufficiently near t , W ( n ) S ( s )( −∞ , a/ ≤ K ( n ) ( s ) . (4.56) DF QUEUES WITH RENEGING Also, for s ∈ [ t − a/ , t ) sufficiently near t so that A ( n ) ( s ) = A ( n ) ( t ) holds[such s exist due to the assumption that △ A ( n ) ( t ) = 0], we have W ( n ) S ( t )( −∞ , ≤ W ( n ) S ( s )( −∞ , a/ . (4.57)The last two relations show that W ( n ) S ( t )( −∞ , ≤ K ( n ) ( t − ), and so, byProposition 4.3, (4.45) holds. Equation (4.46) follows from (4.6), (4.45),(4.39) and the assumption △ A ( n ) ( t ) = 0. Because W ( n ) S ( t ) > W ( n ) S decreases at unit rate in a neighborhood of t [see (2.6)–(2.8)]. In ad-dition, (4.6), (4.45) and the fact that, again by Proposition 4.3, K ( n ) can-not increase on [ σ ( n ) k , τ ( n ) k ] except by a jump and hence is constant in aneighborhood of t , together imply that U ( n ) also decreases at unit rate ina neighborhood of t . Furthermore, the nature of the EDF discipline and(4.55) show that at t , the standard system is serving a customer with leadtime no greater than E ( n ) ( t ). Combining the above properties with thefact that U ( n ) ( t ) | ( E ( n ) ( t ) , ∞ ) = W ( n ) S ( t ) | ( E ( n ) ( t ) , ∞ ) by (4.9), we conclude thatif U ( n ) ( t − ) { E ( n ) ( t − ) } >
0, then U ( n ) ( · ) { E ( n ) ( · ) } decreases with unit rate ina neighborhood of t . On the other hand, if U ( n ) ( t − ) { E ( n ) ( t − ) } = 0, thensince ∆ A ( n ) ( t ) = 0, E ( n ) jumps up at t . Indeed, in this case, W ( n ) S ( t )( −∞ , E ( n ) ( t − )] = W ( n ) S ( t − )( −∞ , E ( n ) ( t − )] = K ( n ) ( t − ) = K ( n ) ( t ) . This means that E ( n ) ( t ) = min { y ∈ R |W ( n ) S ( t )( −∞ , y ] > K ( n ) ( t ) } = min { y > E ( n ) ( t − ) |W ( n ) S ( t ) { y } > } . It follows that W ( n ) S ( t )( E ( n ) ( t − ) , E ( n ) ( t )) = 0 . (4.58)Using the definition of E ( n ) ( t ), (4.46), (4.9), the assumption U ( n ) ( t − ) { E ( n ) ( t − ) } = 0, the assumption △ A ( n ) ( t ) = 0, and (4.58), we obtain U ( n ) ( t )[ E ( n ) ( t ) , ∞ ) = U ( n ) ( t ) = U ( n ) ( t − )= U ( n ) ( t − )[ E ( n ) ( t − ) , ∞ )= U ( n ) ( t − )( E ( n ) ( t − ) , ∞ )= W ( n ) S ( t − )( E ( n ) ( t − ) , ∞ )= W ( n ) S ( t )( E ( n ) ( t − ) , ∞ )= W ( n ) S ( t )[ E ( n ) ( t ) , ∞ ) . KRUK, LEHOCZKY, RAMANAN AND SHREVE
From (4.9) we see now that U ( n ) ( t ) = W ( n ) S ( t ) | [ E ( n ) ( t ) , ∞ ) .(ii) By (4.30), (4.4) and (4.5), for s ∈ ( σ ( n ) k , t ) we have W ( n ) S ( s )( −∞ , E ( n ) ( s )] > K ( n ) ( s ) . (4.59)As s ↑ t in (4.59), by (4.40), (4.38), and the case (ii) assumptions △ A ( n ) ( t ) =0 and E ( n ) ( t − ) = 0, we get W ( n ) S ( t )( −∞ ,
0] = W ( n ) S ( t − )( −∞ , ≥ K ( n ) ( t − ) . (4.60)When combined with Proposition 4.3, this implies K ( n ) ( t ) = K ( n ) ( t − ) ∨ W ( n ) S ( t )( −∞ ,
0] = W ( n ) S ( t )( −∞ , . (4.61)By (4.4) and (4.5), for s ∈ ( σ ( n ) k , t ), U ( n ) ( s ) { E ( n ) ( s ) } = W ( n ) S ( s )( −∞ , E ( n ) ( s )] − K ( n ) ( s ) . Letting s ↑ t , invoking (4.40), (4.60) and (4.61), and recalling the assumption E ( n ) ( t − ) = 0, we obtain U ( n ) ( t − ) { } = W ( n ) S ( t − )( −∞ , − K ( n ) ( t − ) = K ( n ) ( t ) − K ( n ) ( t − ) , (4.62)and the first equality in (4.47) follows. The second equality in (4.47) followsfrom (4.6), (4.39) and the assumption △ A ( n ) ( t ) = 0. Moreover, by (4.5) and(4.61), for every y ∈ R , U ( n ) ( t )( −∞ , y ] = [ W ( n ) S ( t )( −∞ , y ] − W ( n ) S ( t )( −∞ , + (4.63) = W ( n ) S ( t ) | (0 , ∞ ) ( −∞ , y ] . (iii) Let a = E ( n ) ( t − ). We can deduce (4.45) from (4.56) and (4.57) asin (i), with the only difference that now (4.57), for s < t sufficiently closeto t such that A ( n ) ( t − ) = A ( n ) ( s ), follows from the fact that L ( n ) A ( n ) ( t ) > t does not contribute to W ( n ) S ( t )( −∞ , y < a , let ε = ( a − y ) / L ( n ) A ( n ) ( t ) ≥ a > y + ε .Thus, for s < t , s sufficiently close to t [so as to ensure that A ( n ) ( t − ) = A ( n ) ( s )], we have W ( n ) S ( t )( −∞ , y ] ≤ W ( n ) S ( s )( −∞ , y + ε ] ≤ K ( n ) ( s ), where thelast inequality uses (4.5) and the fact that y + ε < E ( n ) ( t − ). Letting s ↑ t ,we obtain W ( n ) S ( t )( −∞ , y ] ≤ K ( n ) ( t − ), which, together with (4.45), showsthat y < E ( n ) ( t ). Thus, E ( n ) ( t − ) ≤ E ( n ) ( t ) or, equivalently, △ E ( n ) ( t ) ≥ U ( n ) ( t − )( −∞ , y ] = [ W ( n ) S ( t − )( −∞ , y ] − K ( n ) ( t − )] + . (4.64) DF QUEUES WITH RENEGING Indeed, for any y such that W ( n ) S ( t − ) { y } = 0, (4.64) follows from (4.5), inwhich t is replaced by s < t , by taking s ↑ t . However, the family of sets( −∞ , y ] with W ( n ) S ( t − ) { y } = 0 forms a separating class in B ( R ), and so(4.64) holds for all y . Moreover, using (4.38), (4.45) and (4.5), we see that U ( n ) ( t )( −∞ , y ](4.65) = [ W ( n ) S ( t − )( −∞ , y ] − K ( n ) ( t − ) + v ( n ) A ( n ) ( t ) δ L ( n ) A ( n )( t ) ( t ) ( −∞ , y ]] + . When combined with (4.64), this shows that U ( n ) ( t )( −∞ , y ] = U ( n ) ( t − )( −∞ , y ] , y < L ( n ) A ( n ) ( t ) ( t ) . (4.66)On the other hand, if y ≥ L ( n ) A ( n ) ( t ) ( t ), then y ≥ E ( n ) ( t − ) and (4.64) becomes U ( n ) ( t − )( −∞ , y ] = W ( n ) S ( t − )( −∞ , y ] − K ( n ) ( t − ) . From (4.65), we now have U ( n ) ( t )( −∞ , y ] = U ( n ) ( t − )( −∞ , y ] + v ( n ) A ( n ) ( t ) , y ≥ L ( n ) A ( n ) ( t ) . (4.67)When combined, (4.66) and (4.67) prove (4.48).(iv) We have L ( n ) A ( n ) ( t ) >
0, and so (4.45) holds by the same argument asin case (iii), but now with a = L ( n ) A ( n ) ( t ) . The assumptions L ( n ) A ( n ) ( t ) < E ( n ) ( t − )and △ A ( n ) ( t ) = 1, along with the relations (4.38), (4.5), (4.45) and the def-inition of E ( n ) , imply that W ( n ) S ( t )( −∞ , E ( n ) ( t − )] = W ( n ) S ( t − )( −∞ , E ( n ) ( t − )] + v ( n ) A ( n ) ( t ) > W ( n ) S ( t − )( −∞ , E ( n ) ( t − )] ≥ K ( n ) ( t − )= K ( n ) ( t ) . Invoking (4.5) again, this shows that U ( n ) ( t )( −∞ , E ( n ) ( t − )] >
0, which im-plies E ( n ) ( t ) ≤ E ( n ) ( t − ). Now, let y < a = L ( n ) A ( n ) ( t ) and let ε = ( a − y ) / y + ε < a < E ( n ) ( t − ) and (4.45), weobtain W ( n ) S ( t )( −∞ , y ] ≤ W ( n ) S ( t − )( −∞ , y + ε ] ≤ K ( n ) ( t − ) = K ( n ) ( t ) . This shows that y < E ( n ) ( t ), which proves (4.49). In addition, by (4.6) and(4.45), we have U ( n ) ( t ) = W ( n ) S ( t ) − K ( n ) ( t ) KRUK, LEHOCZKY, RAMANAN AND SHREVE = W ( n ) S ( t − ) + v ( n ) A ( n ) ( t ) − K ( n ) ( t − )= U ( n ) ( t − ) + v ( n ) A ( n ) ( t ) , and (4.50) follows. Furthermore, since E ( n ) ( t ) ≤ E ( n ) ( t − ) by (4.49), the re-lations (4.9), (4.38) and the assumption L ( n ) A ( n ) ( t ) < E ( n ) ( t − ) imply U ( n ) ( t ) | ( E ( n ) ( t − ) , ∞ ) = W ( n ) S ( t ) | ( E ( n ) ( t − ) , ∞ ) = W ( n ) S ( t − ) | ( E ( n ) ( t − ) , ∞ ) = U ( n ) ( t − ) | ( E ( n ) ( t − ) , ∞ ) . This establishes (4.51).Finally, to prove (4.52), we will consider two cases.
Case I. E ( n ) ( t ) < E ( n ) ( t − ).By (4.9), we know that U ( n ) ( t ) { E ( n ) ( t − ) } = W ( n ) S ( t ) { E ( n ) ( t − ) } . In turn, when combined with (4.64) and the definition of E ( n ) , this showsthat U ( n ) ( t − ) { E ( n ) ( t − ) } = U ( n ) ( t − )( −∞ , E ( n ) ( t − )] − U ( n ) ( t − )( −∞ , E ( n ) ( t − ))= W ( n ) S ( t − )( −∞ , E ( n ) ( t − )] − K ( n ) ( t − ) − [ W ( n ) S ( t − )( −∞ , E ( n ) ( t − )) − K ( n ) ( t − )] + ≤ W ( n ) S ( t − )( −∞ , E ( n ) ( t − )] − W ( n ) S ( t − )( −∞ , E ( n ) ( t − ))= W ( n ) S ( t ) { E ( n ) ( t − ) } = U ( n ) ( t ) { E ( n ) ( t − ) } , and so (4.52) holds. Case
II. E ( n ) ( t ) = E ( n ) ( t − ).By (4.5), (4.38), (4.45), (4.64) and the definition of E ( n ) , U ( n ) ( t ) { E ( n ) ( t ) } = U ( n ) ( t )( −∞ , E ( n ) ( t )]= W ( n ) S ( t )( −∞ , E ( n ) ( t )] − K ( n ) ( t )= W ( n ) S ( t − )( −∞ , E ( n ) ( t − )] + v ( n ) A ( n ) ( t ) − K ( n ) ( t − )= U ( n ) ( t − )( −∞ , E ( n ) ( t − )] + v ( n ) A ( n ) ( t ) DF QUEUES WITH RENEGING = U ( n ) ( t − ) { E ( n ) ( t − ) } + v ( n ) A ( n ) ( t ) , which establishes (4.52) in this case as well. Since E ( n ) ( t ) ≤ E ( n ) ( t − ), thetwo cases above are exhaustive, and so (4.52) is proved.(v) Equation (4.63) holds by the same argument as in (ii), but where nowthe equality in (4.60) follows from the fact that L ( n ) A ( n ) ( t ) >
0. Let U ( n )1 ( t ) ∆ = U ( n ) ( t − ) + v ( n ) A ( n ) ( t ) δ L ( n ) A ( n )( t ) − U ( n ) ( t − ) { } δ . We want to show that U ( n ) ( t ) = U ( n )1 ( t ). By (4.10), U ( n ) ( t ) and U ( n ) ( t − ) are supported on (0 , ∞ ) and [0 , ∞ ),respectively. Thus, U ( n ) ( t )( −∞ , y ] = U ( n )1 ( t )( −∞ , y ] = 0 , y ≤ . (4.68)By (4.9) and the fact that E ( n ) ( t − ) = 0, we have U ( n ) ( t − ) | (0 , ∞ ) = W ( n ) S ( t − ) | (0 , ∞ ) .The last two statements, along with (4.38), (4.63) and another applicationof (4.9), show that U ( n )1 ( t ) | (0 , ∞ ) = U ( n ) ( t − ) | (0 , ∞ ) + v ( n ) A ( n ) ( t ) δ L ( n ) A ( n )( t ) = W ( n ) S ( t − ) | (0 , ∞ ) + v ( n ) A ( n ) ( t ) δ L ( n ) A ( n )( t ) = W ( n ) S ( t ) | (0 , ∞ ) = U ( n ) ( t ) | (0 , ∞ ) . This, together with (4.68), shows that U ( n ) ( t ) = U ( n )1 ( t ). (cid:3) The last result of this section concerns the evolution of U ( n ) . Despitethe different ways in which arriving mass is distributed in the system withreneging and the reference system, in both systems one can keep track of thetotal mass in system by beginning with the arrived mass (which is the samein both systems), subtracting the reduction in mass due to service (whichoccurs continuously at unit rate per unit time whenever mass is present), andsubtracting the mass that has become late and been deleted. In particular,a simple mass balance shows that W ( n ) ( t ) = V ( n ) ( A ( n ) ( t )) − Z t I { W ( n ) ( s ) > } ds − R ( n ) W ( t ) , (4.69)where we recall that R ( n ) W is the total amount of reneged work in the renegingsystem, which admits the representation (2.18), R ( n ) W ( t ) = P } ds − R ( n ) U ( t ) , (4.70)where R ( n ) U ( t ) ∆ = X
For every t ≥ , equation (4.70) holds. Moreover, R ( n ) U = K ( n )+ and hence U ( n ) = N ( n ) + I ( n ) U − K ( n )+ , (4.72) where, for t ≥ , I ( n ) U ( t ) ∆ = Z t I { U ( n ) ( s )=0 } ds. (4.73) Proof.
For t ≥
0, let e U ( n ) ( t ) be equal to the right-hand side of (4.70).By (4.41) of Lemma 4.7, we have U ( n ) (0) = 0 = e U ( n ) (0). Moreover, for every k ≥
1, by Lemma 4.7 and the definition of σ ( n ) k , it follows that U ( n ) ( t − ) = U ( n ) ( t ) = 0 and △ V ( n ) ( A ( n ) ( t )) = 0 for t ∈ ( τ ( n ) k − , σ ( n ) k ), U ( n ) ( σ ( n ) k − ) = 0 and △ U ( n ) ( σ ( n ) k ) = △ V ( n ) ( A ( n ) ( σ nk )). When compared with the right-hand sideof (4.70), this shows that U ( n ) and e U ( n ) are both flat on ( τ ( n ) k − , σ ( n ) k ), with anupward jump at σ ( n ) k of size △ V ( n ) ( A ( n ) ( σ ( n ) k )). Thus, to prove the corollary,it suffices to show that the increments of e U ( n ) and U ( n ) on the intervals( σ ( n ) k , τ ( n ) k ], k ≥
1, coincide.Fix k ≥
1. We first show that∆ e U ( n ) ( τ ( n ) k ) = ∆ U ( n ) ( τ ( n ) k ) . (4.74)Equality (4.30) shows that there cannot be an arrival at time τ ( n ) k , for suchan arrival would have a positive lead time and hence increase W ( n ) S with-out increasing K ( n ) (see Proposition 4.3). In other words, ∆ A ( n ) ( τ ( n ) k ) = 0.Because there is no arrival at τ ( n ) k , the measure-valued process W ( n ) S is con-tinuous at τ ( n ) k . Taking the limit in (4.5) as t ↑ τ ( n ) k , we obtain U ( n ) ( τ ( n ) k − )( −∞ ,
0] = [ W ( n ) S ( τ ( n ) k )( −∞ , − K ( n ) ( τ ( n ) k − )] + = ∆ K ( n ) ( τ ( n ) k ) , DF QUEUES WITH RENEGING where the last equality is a consequence of (4.36). However, (4.10) impliesthat U ( n ) ( τ ( n ) k − )( −∞ , ≤ lim t ↑ τ ( n ) k U ( n ) ( t )( −∞ ,
0) = 0, so ∆ e U ( n ) ( τ ( n ) k ) = −U ( n ) ( τ ( n ) k − ) { } = − ∆ K ( n ) ( τ ( n ) k ). From (4.6) and the continuity of W ( n ) S at τ ( n ) k , we see that − ∆ K ( n ) ( τ ( n ) k ) is also equal to ∆ U ( n ) ( τ ( n ) k ), and (4.74)is proved.We next show that ∆ e U ( n ) ( t ) = ∆ U ( n ) ( t ) for t ∈ ( σ ( n ) k , τ ( n ) k ). If E ( n ) ( t − ) >
0, then the definitions of E ( n ) and e U ( n ) , and statements (i), (iii) and (iv) ofProposition 4.8 show that △ U ( n ) ( t ) = △ e U ( n ) ( t ) = △ V ( n ) ( A ( n ) ( t )) . On the other hand, if E ( n ) ( t − ) = 0, then properties (ii) and (v) of Proposi-tion 4.8 and the definition of e U ( n ) show that △ U ( n ) ( t ) = △ e U ( n ) ( t ) = △ A ( n ) ( t ) v ( n ) A ( n ) ( t ) − U ( n ) ( t − ) { } . Now, let S ( n ) be the (random) set of times s ≥ U ( n ) ( s ) > △ A ( n ) ( s ) > ,E ( n ) ( s − ) = 0or U ( n ) ( s − ) { E ( n ) ( s − ) } = 0 . Suppose U ( n ) ( s ) >
0. If E ( n ) ( s − ) = 0, then the fact that E ( n ) ( s ) > △ E ( n ) ( s ) >
0, while if U ( n ) ( s − ) { E ( n ) ( s − ) } = 0, the defini-tion of E ( n ) ( s ) implies that △ ( U ( n ) ( s ) { E ( n ) ( s ) } ) >
0. Thus, the set S ( n ) is countable, and on the set { s ∈ ( σ ( n ) k , t ) : U ( n ) ( s ) > } \ S ( n ) , the process U ( n ) decreases with unit rate by Proposition 4.8(i). Therefore, the totalamount of this decrease on any time interval of the form ( σ ( n ) k , t ) equals R tσ ( n ) k I { U ( n ) ( s ) > } ds , which coincides with the absolutely continuous part of e U ( n ) ( t ) − e U ( n ) ( σ ( n ) k − ) on the same interval. This concludes the proof of(4.70).Adding and subtracting t to (4.70), by the definition (2.6) of the netputprocess N ( n ) and the nonnegativity of U ( n ) , we obtain U ( n ) ( t ) = N ( n ) ( t ) + Z t I { U ( n ) ( s )=0 } ds − R ( n ) U ( t ) , (4.75)while substituting (4.15) and (2.8) into (4.6), we have U ( n ) ( t ) = N ( n ) ( t ) + I ( n ) S ( t ) + K ( n ) − ( t ) − K ( n )+ ( t ) KRUK, LEHOCZKY, RAMANAN AND SHREVE for t ≥
0. On the other hand, we know that Z [0 , ∞ ) I { U ( n ) ( s ) > } dI ( n ) S ( s ) = 0 and Z [0 , ∞ ) I { U ( n ) ( s ) > } dK ( n ) − ( s ) = 0 , where the former equality holds because W ( n ) S ≥ U ( n ) by (4.8), and I ( n ) S increases only at times when W ( n ) S is zero, while the latter holds by (4.16).From the last three displays, we conclude that Z [0 , ∞ ) I { U ( n ) ( s ) > } dR ( n ) U ( s ) = Z [0 , ∞ ) I { U ( n ) ( s ) > } dK ( n )+ ( s ) . (4.76)On the other hand, since U ( n ) ( s ) = 0 implies ∆ A ( n ) ( s ) = 0, from properties(i) and (ii) of Proposition 4.8 and the fact that R ( n ) U is a pure jump processwith ∆ R ( n ) U ( t ) = U ( n ) ( t − ) { } , it follows that Z [0 , ∞ ) I { U ( n ) ( s )=0 } dR ( n ) U ( s ) = Z [0 , ∞ ) I { U ( n ) ( s )=0 } dK ( n )+ ( s ) . Together, the last two equalities imply R ( n ) U = K ( n )+ , which, when substitutedinto (4.70), yields (4.72). (cid:3)
5. The reneging system.
In this section we bound the difference in work-load between the pre-limit reference and reneging systems—Lemma 5.2 pro-vides a lower bound, while Lemma 5.6 provides an upper bound. The proofof the upper bound uses an optimality property of EDF that may be ofindependent interest.
Theorem 5.1.
Let π be a service policy for a single-station, single-customer-class queueing system with reneging such that the customer arrivaltimes to this system do not have a finite accumulation point. Let R π ( t ) bethe amount of work removed from this system up to time t due to lateness.Let R W ( t ) be the amount of work removed due to lateness up to time t fromthe EDF system with reneging and the same interarrival times, service timesand lead times as in the former system. Then for every t ≥ , we have R W ( t ) ≤ R π ( t ) . (5.1)The proof of Theorem 5.1 is deferred to the Appendix. The related factthat the EDF protocol minimizes the number of late customers in the G/M/c queue was proved in [29], and the main idea of our proof is similar to thatof [29]. However, our argument is pathwise and the only assumption onthe distribution of the system stochastic primitives that we impose is thatcustomer arrivals do not have a finite accumulation point. This assumptionis clearly satisfied almost surely by a
GI/G/
DF QUEUES WITH RENEGING Comparison results.
In this section, we establish bounds on the dif-ference between the processes U ( n ) and W ( n ) . In Section 6.1, this differencewill be shown to be negligible in the heavy traffic limit. We start with Lemma5.2 showing that W ( n ) ≤ U ( n ) , which implies that R ( n ) U ≤ R ( n ) W (see Corollary5.3).In the proofs of these results, we will make frequent use of the observationthat, by (4.69) and (4.70), W ( n ) ( t ) − U ( n ) ( t ) = Z t I { U ( n ) ( s ) > } ds − Z t I { W ( n ) ( s ) > } ds (5.2) + R ( n ) U ( t ) − R ( n ) W ( t )for t ∈ [0 , ∞ ). Lemma 5.2.
For every t ≥ , we have W ( n ) ( t ) ≤ U ( n ) ( t ) . (5.3) Proof.
Let τ ∆ = min { t ≥ W ( n ) ( t ) > U ( n ) ( t ) } . (5.4)If τ = + ∞ , then (5.3) holds. Assume τ < + ∞ . In this case, we claim thatthe minimum on the right-hand side of (5.4) is attained. Indeed, (5.2) andthe fact that R ( n ) U and R ( n ) W are pure jump processes show that the only waythat W ( n ) − U ( n ) can become strictly positive is via a jump. Thus W ( n ) ( τ ) >U ( n ) ( τ ). Since W ( τ )( −∞ ,
0] = U ( τ )( −∞ ,
0] = 0 (in fact, this equality holdsfor any time t ), this means there must exist a y > W ( n ) ( τ )( y, ∞ ) > U ( n ) ( τ )( y, ∞ ) . (5.5)Let τ = inf { t ∈ [0 , τ ] : W ( n ) ( t )( y + τ − t, ∞ ) > U ( n ) ( t )( y + τ − t, ∞ ) } . (5.6)By (5.5), the above infimum is over a nonempty set. Lemma 4.7 and Propo-sition 4.8 imply that the only difference in the dynamics of W ( n ) and U ( n ) is that the arriving mass v ( n ) k is concentrated at L ( n ) k in the case of the EDFsystem with reneging and distributed in [ L ( n ) k , ∞ ) in the reference system.On the other hand, in both systems at time t ∈ [0 , τ ], no mass leaves theinterval ( y + τ − t, ∞ ) due to lateness. This implies that W ( n ) ( t )( y + τ − t, ∞ ) − U ( n ) ( t )( y + τ − t, ∞ ), t ∈ [0 , τ ], has no positive jumps and therefore W ( n ) ( t )( y + τ − τ , ∞ ) = U ( n ) ( t )( y + τ − τ , ∞ ) . (5.7) KRUK, LEHOCZKY, RAMANAN AND SHREVE
By (5.5) and (5.7), τ < τ . Thus, there exists t ∈ ( τ , τ ), where t − τ isarbitrarily small and W ( n ) ( t )( y + τ − t, ∞ ) > U ( n ) ( t )( y + τ − t, ∞ ) . (5.8)However, we claim that (5.7) and (5.8) imply that for all t ∈ ( τ , τ ), where t − τ is small enough, it must be that W ( n ) ( t )(0 , y + τ − t ] > , (5.9) U ( n ) ( t )(0 , y + τ − t ] = 0 . (5.10)Indeed, if (5.9) is false, then the left-hand side of (5.8) is equal to W ( n ) ( t ),and consequently decreases with unit speed as long as it is nonzero in sometime interval beginning with τ . Similarly, if (5.10) is false, the right-handside of (5.8) is constant on some interval beginning with τ . In both cases,due to (5.7), (5.8) cannot hold for t ∈ ( τ , τ ) with t − τ arbitrarily small.But (5.8)–(5.10) yield W ( n ) ( t ) > U ( n ) ( t ) for some t < τ , which contradicts(5.4). (cid:3) Corollary 5.3.
For every t ≥ , R ( n ) U ( t ) ≤ R ( n ) W ( t ) . (5.11) Moreover, for k ≥ and t ≥ σ ( n ) k , R ( n ) U ( t ) − R ( n ) U ( σ ( n ) k − ) ≤ R ( n ) W ( t ) − R ( n ) W ( σ ( n ) k − ) . (5.12) Proof.
Lemma 5.2 and (5.2) imply that for 0 ≤ s ≤ t ,( R ( n ) U ( t ) − R ( n ) U ( s )) − ( R ( n ) W ( t ) − R ( n ) W ( s )) ≤ U ( n ) ( s ) − W ( n ) ( s ) . (5.13)Substituting s = 0 into (5.13) and using the fact that R ( n ) U (0) = R ( n ) W (0) = U ( n ) (0) = W ( n ) (0) = 0, we obtain (5.11). Likewise, for 0 ≤ s ≤ σ ( n ) k ≤ t , tak-ing limits as s tends to σ ( n ) k in (5.13), and using the fact that U ( n ) ( σ ( n ) k − ) = W ( n ) ( σ ( n ) k − ) = 0, which follows from (4.41), Lemma 5.2 and the nonnega-tivity of W ( n ) , we obtain (5.12). (cid:3) The proofs of Lemma 5.2 and Corollary 5.3 show the following moregeneral (and intuitively obvious) fact: if all customers in the EDF systemwith reneging get larger deadlines, this results in a larger workload at everytime t and a smaller total amount of mass removed from the system due tolateness in the time interval [0 , t ].We now establish an inequality between the frontiers in both systems. DF QUEUES WITH RENEGING Lemma 5.4.
For every t ≥ such that U ( n ) ( t ) > , we have E ( n ) ( t ) ≤ F ( n ) ( t ) . (5.14) Proof.
Subtracting (4.5) from (4.6), we see that for any y ∈ R , U ( n ) ( t )( y, ∞ ) = W ( n ) S ( t ) − K ( n ) ( t ) − [ W ( n ) S ( t )( −∞ , y ] − K ( n ) ( t )] + (5.15) ≤ W ( n ) S ( t )( y, ∞ ) . Now, assume that for some t we have F ( n ) ( t ) < E ( n ) ( t ). In this case, W ( n ) ( t ) ≥ W ( n ) ( t ) { C ( n ) ( t ) } + W ( n ) ( t )( F ( n ) ( t ) , ∞ )= W ( n ) ( t ) { C ( n ) ( t ) } + V ( n ) ( t )( F ( n ) ( t ) , ∞ ) ≥ W ( n ) ( t ) { C ( n ) ( t ) } + V ( n ) ( t )[ E ( n ) ( t ) , ∞ )(5.16) ≥ W ( n ) ( t ) { C ( n ) ( t ) } + W ( n ) S ( t )[ E ( n ) ( t ) , ∞ ) ≥ W ( n ) ( t ) { C ( n ) ( t ) } + U ( n ) ( t )[ E ( n ) ( t ) , ∞ ) ≥ U ( n ) ( t ) , where the second line follows from the fact that none of the customers inthe EDF system with reneging that have lead times at time t greater than F ( n ) ( t ) has received any service up to time t , the second-last inequalityfollows from (5.15) and the last line holds due to the equality U ( n ) ( t ) = U ( n ) ( t )[ E ( n ) ( t ) , ∞ ). When combined with the assumption that U ( n ) ( t ) > W ( n ) ( t ) >
0. This, in turn, implies that W ( n ) ( t ) { C ( n ) ( t ) } > (cid:3) Let D ( n ) ( t ) be the amount of work deleted by the EDF system withreneging in the time interval [0 , t ] that is associated with customers whoselead times upon arrival were smaller than the value of the frontier at thetime of their arrival. In the proof of the next lemma, we will make use ofthe elementary fact that by the definition of F ( n ) we have F ( n ) ( t ) − ( t − t ) ≤ F ( n ) ( t ) , S ( n )1 ≤ t ≤ t . (5.17) Lemma 5.5.
For every t ≥ , U ( n ) ( t ) − W ( n ) ( t ) ≤ D ( n ) ( t ) . (5.18) KRUK, LEHOCZKY, RAMANAN AND SHREVE
Proof. If t ∈ [ τ ( n ) k − , σ ( n ) k ) for some k ≥
1, then U ( n ) ( t ) = 0 by (4.41).Thus, by (4.19), it suffices to prove (5.18) on [ σ ( n ) k , τ ( n ) k ) for every k ≥
1. Let k ≥
1. Suppose that (5.18) is false for some t ∈ [ σ ( n ) k , τ ( n ) k ). Let τ ∆ = min { t ∈ [ σ ( n ) k , τ ( n ) k ) | U ( n ) ( t ) − W ( n ) ( t ) > D ( n ) ( t ) } . (5.19)We first argue that the minimum on the right-hand side of (5.19) is attained.Indeed, by (5.2) and Lemma 5.2, it is clear that U ( n ) − W ( n ) cannot increaseexcept by a jump that is due to lateness in the EDF system with reneging.Thus, we have W ( n ) ( τ − ) { } > U ( n ) ( τ ) − W ( n ) ( τ ) > D ( n ) ( τ ) . (5.20)Also, (4.41), (4.42) and Lemma 5.2 imply that U ( n ) ( σ ( n ) k ) = △ U ( n ) ( σ ( n ) k ) = △ W ( n ) ( σ ( n ) k ) = W ( n ) ( σ ( n ) k ), so σ ( n ) k < τ . In particular, (5.19) implies U ( n ) ( τ − ) − W ( n ) ( τ − ) ≤ D ( n ) ( τ − ) . (5.21)Let k be the index of the customer arriving at time σ ( n ) k , that is, S ( n ) k = σ ( n ) k . Let k ≥ k be the index of a customer who reneges in the renegingsystem at time τ . There must be such a customer, and there may in fact bemore than one such customer. The amount of work associated with all suchcustomers at time τ is W ( n ) ( τ − ) { } , and we seek to show that this work isbounded above by △ D ( n ) ( τ ). We have S ( n ) k ∈ [ σ ( n ) k , τ ) and L ( n ) k − ( τ − S ( n ) k ) =0. The subsequent analysis is divided into two cases. Case
I. For every customer k chosen as just described, assume there is acustomer ℓ arriving in the time interval [ σ ( n ) k , S ( n ) k ] who is at least as urgent ascustomer k when customer k arrives but whose associated mass in the ref-erence system is at least partly assigned so that upon the arrival of customer k , this mass is to the right of L ( n ) k . In other words, ℓ ∈ [ k , k ], L ( n ) ℓ − ( S ( n ) k − S ( n ) ℓ ) ≤ L ( n ) k and △W ( n ) ( S ( n ) ℓ ) { L ( n ) ℓ } > △U ( n ) ( S ( n ) ℓ )[ L ( n ) l , L ( n ) k + S ( n ) k − S ( n ) ℓ ].In this case, △U ( n ) ( S ( n ) ℓ )( L ( n ) k + S ( n ) k − S ( n ) ℓ , ∞ ) >
0. Indeed, by Lemma 4.7and Proposition 4.8(iv) (describing the only case in which part of the massof a new customer is distributed by the reference workload to a point otherthan its lead time) △ U ( S ( n ) ℓ ) = v ( n ) ℓ and △U ( n ) ( S ( n ) ℓ )( −∞ , L ( n ) ℓ ) = 0 [see(4.42), (4.43), (4.49), (4.50) and (4.4)]. Let s > L ( n ) k + S ( n ) k − S ( n ) ℓ satisfy △U ( n ) ( S ( n ) ℓ ) { s } >
0. Such a point s exists since the measure U ( n ) ( S ( n ) ℓ ) isdiscrete.If ℓ > k (e.g., ℓ = k ), then, by (4.51) in Proposition 4.8(iv) and Lemma5.4, we have s ≤ E ( n ) ( S ( n ) ℓ − ) ≤ F ( n ) ( S ( n ) ℓ − ) ≤ F ( n ) ( S ( n ) ℓ ). Thus, by (5.17), L ( n ) k < s − ( S ( n ) k − S ( n ) ℓ ) ≤ F ( n ) ( S ( n ) ℓ ) − ( S ( n ) k − S ( n ) ℓ ) ≤ F ( n ) ( S ( n ) k ). DF QUEUES WITH RENEGING If ℓ = k , then, because U ( n ) ( S ( n ) k ) { s } >
0, we have W ( n ) S ( S ( n ) k ) { s } > U ( n ) . In this case W ( n ) ( S ( n ) k ) { s } = 0, because W ( n ) ≡ τ ( n ) k − , σ ( n ) k ) by (4.41) and Lemma 5.2, so W ( n ) ( S ( n ) k ) = W ( n ) ( σ ( n ) k ) = v ( n ) k δ L ( n ) k and s > L ( n ) k + S ( n ) k − S ( n ) k ≥ L ( n ) k by the definitions of ℓ and s . Thus, acustomer with lead time equal to s at time S ( n ) k has already been in servicein the EDF system with reneging, so L ( n ) k + S ( n ) k − S ( n ) k < s ≤ F ( n ) ( S ( n ) k ) andconsequently, by (5.17), L ( n ) k < F ( n ) ( S ( n ) k ) − ( S ( n ) k − S ( n ) k ) ≤ F ( n ) ( S ( n ) k ).Thus, regardless of the value of ℓ , L ( n ) k < F ( n ) ( S ( n ) k ). In other words, un-der the Case I assumption, every customer k who becomes late at time τ in the EDF system with reneging arrived with initial lead time smaller thanthe value of F ( n ) at the time of its arrival. The work associated with thesecustomers deleted at time τ is △ D ( n ) ( τ ). We conclude that W ( n ) ( τ − ) { } = △ D ( n ) ( τ ). However, by (5.2), we have △ ( U ( n ) − W ( n ) )( τ ) ≤ W ( n ) ( τ − ) { } ,and so △ ( U ( n ) − W ( n ) )( τ ) ≤ △ D ( n ) ( τ ). This, together with (5.21), contra-dicts (5.20). Case
II. For a customer k chosen as described above, assume that everycustomer ℓ arriving in the time interval [ σ ( n ) k , S ( n ) k ] who is as least as urgentas customer k when customer k arrives has all its associated mass ini-tially assigned in the reference system to the interval (0 , L ( n ) k + S ( n ) k − S ( n ) ℓ ]upon arrival. Customers ℓ who are less urgent then k must have lead timessatisfying L ( n ) ℓ > L ( n ) k + S ( n ) k − S ( n ) ℓ , and hence the mass brought by suchcustomers must be initially assigned to the half-line ( L ( n ) k + S ( n ) k − S ( n ) ℓ , ∞ )in both systems. Then for every t ∈ [ σ ( n ) k , S ( n ) k ], we have W ( n ) ( t )(0 , L ( n ) k − ( t − S ( n ) k )] ≤ U ( n ) ( t )(0 , L ( n ) k − ( t − S ( n ) k )] , (5.22)as we now explain. Under the Case II assumption the arrival of new massis the same on both sides of (5.22). Furthermore, disregarding lateness andnew arrivals, both sides of (5.22) decrease at unit rate so long as they arenonzero. Finally, by (5.12) the amount of late work removed from the EDFsystem with reneging in the time interval [ σ ( n ) k , t ] is greater than or equal tothe amount of late work removed from U ( n ) in this time interval. Therefore,(5.22) holds for every t ∈ [ σ ( n ) k , S ( n ) k ].We claim that (5.22) in fact holds for all t ∈ [ σ ( n ) k , τ ). Suppose this is notthe case. Let η ∆ = inf { t ∈ [ S ( n ) k , τ ) |W ( n ) ( t )(0 , L ( n ) k − ( t − S ( n ) k )](5.23) > U ( n ) ( t )(0 , L ( n ) k − ( t − S ( n ) k )] } . KRUK, LEHOCZKY, RAMANAN AND SHREVE
The strict inequality in (5.23) can occur only because of an arrival at time t which brings mass to the interval (0 , L ( n ) k − ( t − S ( n ) k )] under the W ( n ) mea-sure but not under the U ( n ) measure. The arrival at time k does not havethis property because the Case II assumption applies to ℓ = k . Therefore, η > S ( n ) k .Also, for t ∈ [ S ( n ) k , τ ), W ( n ) ( t ) { L ( n ) k − ( t − S ( n ) k ) } > , (5.24)because the customer k is present in the EDF system with reneging at time t . By (4.4), (5.24) and the definition of η , we have E ( n ) ( t ) ≤ L ( n ) k − ( t − S ( n ) k )for t ∈ [ S ( n ) k , η ). Thus, E ( n ) ( t − ) ≤ L ( n ) k − ( t − S ( n ) k ) for t ∈ ( S ( n ) k , η ]. We arguethat this implies that the amounts of mass arriving in both the EDF systemwith reneging and the reference workload at any time t ∈ ( S ( n ) k , η ] with leadtimes upon arrival less than or equal to L ( n ) k − ( t − S ( n ) k ) are the same. Indeed,Proposition 4.8, especially (4.51), implies that no mass arriving at time t with lead time smaller than E ( n ) ( t − ) in the EDF system with reneging isdistributed to lead times greater than E ( n ) ( t − ) by the reference workload.Also, Proposition 4.8(iii) and (v) imply that the mass arriving at time t withlead time greater than or equal to E ( n ) ( t − ) is distributed in the same wayby the EDF system with reneging and the reference system. By the sameargument as in the case of t ∈ [ σ ( n ) k , S ( n ) k ], we conclude that (5.22) holds for t ∈ [ S ( n ) k , η ], which contradicts the definition of η . We have shown that (5.22)holds for t ∈ [ σ ( n ) k , τ ).Letting t ↑ τ in (5.22) and using the fact that L ( n ) k − ( τ − S ( n ) k ) = 0,we get W ( n ) ( τ − ) { } ≤ U ( n ) ( τ − ) { } . Thus, by (5.2), △ ( U ( n ) − W ( n ) )( τ ) = W ( n ) ( τ − ) { } − U ( n ) ( τ − ) { } ≤ D ( n ) is nondecreasing, contradicts (5.20). (cid:3) For the sake of the next proof, we define a sequence of auxiliary hybrid sys-tems (with the same stochastic primitives as in the case of the EDF systemsdescribed in Section 2.2) as follows. The hybrid system gives priority to thejobs whose lead times upon arrival are smaller than the current frontier F ( n ) in the corresponding EDF system with reneging. In other words, for each k ,the k th customer arriving at the hybrid system joins the high-priority classif and only if L ( n ) k < F ( n ) ( S ( n ) k ) . (5.25)The system processes high-priority customers according to the FIFO servicediscipline. When the priority class empties, the system goes idle until either DF QUEUES WITH RENEGING another high-priority customer arrives and the system resumes service in themanner described above, or the corresponding EDF system with renegingfinishes serving the customers who have received priority in the hybrid sys-tem. Here, we are using the fact that the high-priority customers leave thehybrid system before they leave the EDF system with reneging, which is aconsequence of the optimality of the EDF discipline established in Theorem5.1. (We have slightly abused the terminology here, identifying the k th cus-tomer in the hybrid system with the corresponding customer from the EDFsystem with reneging, while, formally, only the random variables u ( n ) k , v ( n ) k and L ( n ) k associated with these customers are the same.) Whenever the EDFsystem with reneging finishes serving a batch of customers who have receivedhigh priority in the hybrid system, both systems then serve the low-priorityclass using the EDF discipline until the next high-priority customer arrives.In both systems, if a customer is present when his deadline passes, he leavesthe queue immediately, regardless of his class. The measure-valued workloadprocess associated with the hybrid system will be denoted by W ( n ) H . Lemma 5.6.
For every t ≥ , we have U ( n ) ( t ) − W ( n ) ( t ) ≤ A ( n ) ( t ) X k =1 v ( n ) k ∧ ( W ( n ) ( S ( n ) k − )(0 , F ( n ) ( S ( n ) k )) + v ( n ) k − L ( n ) k ) + (5.26) × I { L ( n ) k By Lemma 5.5, it suffices to show that D ( n ) ( t ) is not greaterthan the right-hand side of (5.26). By Theorem 5.1, D ( n ) ( t ), the amountof unfinished work associated with customers who arrived with lead timessmaller than F ( n ) and were deleted in the time interval [0 , t ] by the EDFsystem with reneging, is not greater than the unfinished work associated withthese customers and deleted by the corresponding hybrid system. Note thatthe customers with lead times satisfying (5.25) form a priority class in boththe EDF system with reneging and the hybrid system, and so their serviceis not affected by the presence of other customers. Furthermore, unfinishedwork associated with deleted customers who arrived with lead times greaterthan or equal to F ( n ) is the same in both systems.For each k , if (5.25) holds, then the k th customer of the hybrid sys-tem belongs to the high-priority class. Moreover, if, for some l < k , L ( n ) l 6. Heavy traffic analysis. In Sections 6.1 and 6.2, respectively, we iden-tify the heavy traffic limit of the scaled workload and the scaled renegedwork in the reneging system. In both cases, this is done by first consideringthe reference system, which is easier to analyze, and then using the boundsderived in Section 5.1 to show that the limits in both systems coincide. Forthe heavy traffic analysis of the reference system, we will find it useful tointroduce the following scaled quantities: b U ( n ) ( t ) ∆ = 1 √ n U ( n ) ( nt ) , b R ( n ) U ( t ) ∆ = 1 √ n R ( n ) U ( nt ) , (6.1) b K ( n )+ ( t ) ∆ = 1 √ n K ( n )+ ( nt ) , and, for every Borel set B ⊂ R , b U ( n ) ( t )( B ) ∆ = 1 √ n U ( n ) ( nt )( √ nB ) . (6.2)Also, define U ∗ ∆ = Φ( W ∗ S ) and U ∗ ( · ) ∆ = U ∗ ( · )( R ) = Φ( W ∗ S )( R ) . (6.3)6.1. Proofs of main results concerning the workload. Proof of Theorem 3.4. In Lemma 6.1, we use the continuity prop-erty of the mapping Φ established in Lemma 4.1, along with the character- DF QUEUES WITH RENEGING ization of the heavy traffic limit of the workload measure-valued process inthe standard system, to identify the heavy traffic limit of the workload inthe reference system. Let Λ H (0) : D [0 , ∞ ) → D [0 , ∞ ) be the mapping defined,for every φ ∈ D [0 , ∞ ) and t ≥ 0, byΛ H (0) ( φ )( t ) ∆ = φ ( t ) − sup s ∈ [0 ,t ] h ( φ ( s ) − H (0)) + ∧ inf u ∈ [ s,t ] φ ( u ) i . (6.4)If φ is nonnegative, then by Theorem 1.4 from [25], Λ H (0) ( φ ) is the functionin D [0 , ∞ ) obtained by double reflection of φ at 0 and H (0). In other words,Λ H (0) ( φ ) takes values in [0 , H (0)] and has the unique decompositionΛ H (0) ( φ ) = φ − κ + + κ − , (6.5)where κ ± are nondecreasing RCLL functions satisfying κ ± (0 − ) = 0 and Z [0 , ∞ ) I { Λ H (0) ( φ )( s ) The process U ∗ satisfies U ∗ = Λ H (0) ( W ∗ S )(6.7) and has the same distribution as W ∗ . Moreover, b U ( n ) ⇒ U ∗ = Φ( W ∗ S ) and b U ( n ) ⇒ W ∗ as n → ∞ . Proof. By the definition of U ∗ and Φ given in (6.3) and (4.2), respec-tively, U ∗ ( t ) = Φ( W ∗ S )( R )( t ) = W ∗ S ( t ) − sup s ∈ [0 ,t ] h W ∗ S ( −∞ , ∧ inf u ∈ [ s,t ] W ∗ S ( u ) i , t ≥ . Since (3.1)–(3.3) imply W ∗ S ( t )( −∞ , 0] = ( W ∗ S ( t ) − H (0)) + for every t ≥ 0, thisshows that U ∗ = Λ H (0) ( W ∗ S ). By the characterization of W ∗ S given at the endof Section 2.4, Λ H (0) ( W ∗ S ) is a Brownian motion with variance ( α + β ) λ per unit time and drift − γ , reflected at 0 and H (0). This proves the firstclaim.Next, using the definition U ( n ) = Φ( W ( n ) S ) and the scaling properties ofΦ, it is easy to see that b U ( n ) = Φ( c W ( n ) S ) . Since, by Theorem 3.2, we knowthat c W ( n ) S ⇒ W ∗ S , where W ∗ S is continuous and W ∗ S ( t ) has a continuous dis-tribution for every t , an application of the continuous mapping theorem,along with the continuity property of Φ stated in Lemma 4.1, shows that KRUK, LEHOCZKY, RAMANAN AND SHREVE b U ( n ) ⇒ Φ( W ∗ S ). This, in particular, implies that b U ( n ) = b U ( n ) ( R ) ⇒ U ∗ . Since U ∗ has the same distribution as W ∗ , this proves the lemma. (cid:3) We identify the heavy traffic limit of the workload in the reneging system.We start with Proposition 6.2, which states that the number of customers inthe EDF system with reneging having lead times not greater than the currentfrontier and the work associated with these customers are negligible underheavy traffic scaling. Then, in Corollary 6.3, we use the comparison resultsestablished in Section 5.1 to show that the workloads in the reference andreneging systems are equal with high probability and so their heavy trafficlimits coincide. Proposition 6.2. The processes c W ( n ) (0 , b F ( n ) ] and b Q ( n ) (0 , b F ( n ) ] con-verge in distribution to zero as n → ∞ . This result holds for the same reason that state-space collapse occursfor priority queues, an idea that can be traced back to [35]. Specifically,in our model, due to the nature of the EDF service discipline, the entirecapacity of the server is always devoted to work that lies to the left or at thefrontier, as long as the system is nonempty. Thus the process W ( n ) (0 , F ( n ) ]is equal to the workload in a single-server GI/G/ V ( n ) ( t )( −∞ , F ( n ) ( t )] − t , t ≥ 0. By showing that F ( n ) ( t ) < √ ny ∗ ,one shows that this (high-priority) queue is in light traffic as n → ∞ , andso its diffusion scaling vanishes in the limit. Since a rigorous proof that c W ( n ) [ b C ( n ) , b F ( n ) ] ⇒ b Q ( n ) [ b C ( n ) , b F ( n ) ] ⇒ c W ( n ) (0 , b C ( n ) ) = b Q ( n ) (0 , b C ( n ) ) = 0 by definition. Corollary 6.3. Let T > . As n → ∞ , P [ U ( n ) ( t ) = W ( n ) ( t ) , ≤ t ≤ nT ] → . (6.8) Proof. Because customers with strictly positive lead times do not re-nege, we have W ( n ) ( S ( n ) k − )(0 , F ( n ) ( S ( n ) k )) ≤ W ( n ) ( S ( n ) k )(0 , F ( n ) ( S ( n ) k )) for k ≥ 1. Thus, by Lemmas 5.2 and 5.6, to prove (6.8), it suffices to show that as n → ∞ , P [ W ( n ) ( S ( n ) k )(0 , F ( n ) ( S ( n ) k )) + v ( n ) k ≤ L ( n ) k , ≤ k ≤ A ( n ) ( nT )] → . However, this follows from the fact that, by (2.15),max ≤ k ≤ A ( n ) ( nT ) v ( n ) k = √ n max ≤ t ≤ T △ b N ( n ) S ( t ) = o ( √ n ) , the inequalities L ( n ) k ≥ √ ny ∗ , y ∗ > 0, and Proposition 6.2. (cid:3) Theorem 3.4 now follows immediately from Lemma 6.1 and Corollary 6.3. DF QUEUES WITH RENEGING Proofs of Proposition 3.5 and Theorem 3.6. We present the proofsof the remaining two limit theorems concerning the measure-valued work-load processes. For this, we need two preliminary results. The first, Lemma6.4, is that the frontier in the reneging system is strictly positive with highprobability. The second result, Proposition 6.5, is a recap of a result estab-lished in [7]. Lemma 6.4. Let T > . As n → ∞ , P [ F ( n ) ( t ) > , ≤ t ≤ nT ] → . (6.9) Proof. Let 0 ≤ t ≤ nT . If W ( n ) ( t ) > 0, then F ( n ) ( t ) is not smaller thanthe lead time of the currently served customer, so F ( n ) ( t ) > 0. If W ( n ) ( t ) = 0,then the customer indexed by A ( n ) ( t ) has already been in service, so F ( n ) ( t ) ≥ L ( n ) A ( n ) ( t ) − ( t − S ( n ) A ( n ) ( t ) ) ≥ √ ny ∗ − u ( n ) A ( n ) ( t )+1 (6.10) ≥ √ ny ∗ − max ≤ k ≤ A ( n ) ( nT )+1 u ( n ) k . However, max ≤ k ≤ A ( n ) ( nT )+1 u ( n ) k = o ( √ n ) by (2.13) (in particular, by thefact that S ∗ has continuous sample paths), so (6.10) implies (6.9). (cid:3) Proposition 6.5 (Proposition 3.4 [7]). Let −∞ < y < y ∗ and T > begiven. As n → ∞ , sup y ≤ y ≤ y ∗ sup ≤ t ≤ T | b V ( n ) ( t )( y, ∞ ) + H ( y + √ nt ) − H ( y ) | P −→ , sup y ≤ y ≤ y ∗ sup ≤ t ≤ T | b A ( n ) ( t )( y, ∞ ) + λH ( y + √ nt ) − λH ( y ) | P −→ . Proof of Proposition 3.5. Let T > 0. We will show that b F ( n ) ⇒ F ∗ in D R [0 , T ]. By definition, y ∗ − √ nt ≤ b F ( n ) ( t ) ≤ y ∗ . Thus, by Proposition 6.5and the fact that H ( y ) = 0 for y ≥ y ∗ ,sup ≤ y ≤ y ∗ sup ≤ t ≤ T | b V ( n ) ( t )( b F ( n ) ( t ) ∨ y, ∞ ) − H ( b F ( n ) ( t ) ∨ y ) | P −→ . (6.11)Putting y = 0 in (6.11) and using Lemma 6.4, we obtainsup ≤ t ≤ T | b V ( n ) ( t )( b F ( n ) ( t ) , ∞ ) − H ( b F ( n ) ( t )) | P −→ . (6.12) KRUK, LEHOCZKY, RAMANAN AND SHREVE For any t ≥ c W ( n ) ( t ) = c W ( n ) ( t )(0 , b F ( n ) ( t )] + c W ( n ) ( t )( b F ( n ) ( t ) , ∞ )(6.13) = c W ( n ) ( t )(0 , b F ( n ) ( t )] + b V ( n ) ( t )( b F ( n ) ( t ) , ∞ ) , where the second line follows from the fact that none of the customersin the EDF system with reneging with lead times at time t greater than F ( n ) ( t ) has received any service up to time t . This, together with Proposi-tion 6.2 and Theorem 3.4, yields b V ( n ) ( b F ( n ) , ∞ ) ⇒ W ∗ . Thus, by (6.12), wehave H ( b F ( n ) ) ⇒ W ∗ in D R [0 , T ]. Applying the continuous function H − toboth sides of this relation and using (3.4), we obtain b F ( n ) ⇒ F ∗ in D R [0 , T ]. (cid:3) Proof of Theorem 3.6. Define a mapping ψ : R → M by the formula ψ ( x )( B ) ∆ = R B ∩ [ x, ∞ ) (1 − G ( η )) dη for x ∈ R and B ∈ B ( R ). It is easy to seethat ψ is continuous. Hence, by Proposition 3.5,( ψ ( b F ( n ) ) , λψ ( b F ( n ) )) ⇒ ( ψ ( F ∗ ) , λψ ( F ∗ )) = ( W ∗ , Q ∗ ) . (6.14)Let T > 0. We claim thatsup ≤ y ≤ y ∗ sup ≤ t ≤ T | c W ( n ) ( t )( y, ∞ ) − ψ ( b F ( n ) ( t ))( y, ∞ ) | P −→ , (6.15) sup ≤ y ≤ y ∗ sup ≤ t ≤ T | b Q ( n ) ( t )( y, ∞ ) − λψ ( b F ( n ) ( t ))( y, ∞ ) | P −→ . (6.16)Indeed, reasoning as in (6.13), we see that, for 0 ≤ y ≤ y ∗ and 0 ≤ t ≤ T , | c W ( n ) ( t )( y, ∞ ) − H ( b F ( n ) ( t ) ∨ y ) |≤ | c W ( n ) ( t )( b F ( n ) ( t ) ∨ y, ∞ ) − H ( b F ( n ) ( t ) ∨ y ) | + c W ( n ) ( t )(0 , b F ( n ) ( t )]= | b V ( n ) ( t )( b F ( n ) ( t ) ∨ y, ∞ ) − ψ ( F ( n ) ( t ))( y, ∞ ) | + c W ( n ) ( t )(0 , b F ( n ) ( t )] . Therefore, (6.15) follows from (6.11) and Proposition 6.2. A similar argumentgives (6.16). We have c W ( n ) ( t )( −∞ , 0] = c W ( n ) ( t )( y ∗ , ∞ ) = b Q ( n ) ( t )( −∞ , 0] = b Q ( n ) ( t )( y ∗ , ∞ ) = 0 and, by Lemma 6.4, P [ ψ ( b F ( n ) ( t ))( −∞ , 0] = 0 , ≤ t ≤ T ] → n → ∞ . Also, ψ ( x )( y ∗ , ∞ ) = 0 for every x ∈ R . Thus, (6.14)–(6.16)imply that ( c W ( n ) , b Q ( n ) ) ⇒ ( W ∗ , Q ∗ ) in D M [0 , T ]. (cid:3) The heavy traffic limit of the reneged work process. In this section,we identify the limit of the sequence { b R ( n ) W , n ∈ N } , thereby proving Theorem3.8. To do this, it is convenient to show that many of the processes underconsideration can be put on a common probability space so that certainweak limits established earlier can be replaced by almost sure limits. DF QUEUES WITH RENEGING Lemma 6.6. The processes c W ( n ) S , b U ( n ) , c W ( n ) , n ∈ N , W ∗ S , U ∗ and W ∗ can be defined on a common probability space (Ω , F , P ) such that P almostsurely, as n → ∞ , c W ( n ) S → W ∗ S , (6.17) c W ( n ) S → W ∗ S , (6.18) c W ( n ) S ( · )( −∞ , → W ∗ S ( · )( −∞ , 0] = ( W ∗ S ( · ) − H (0)) + , (6.19) b U ( n ) → U ∗ (6.20) and c W ( n ) → W ∗ ∆ = U ∗ , (6.21) where c W ( n ) S = c W ( n ) S ( R ) , W ∗ S = W ∗ S ( R ) , b U ( n ) = b U ( n ) ( R ) and U ∗ = U ∗ ( R ) . Fur-thermore, W ∗ S is a Brownian motion with variance ( α + β ) λ per unit timeand drift − γ , reflected at , while U ∗ is a doubly reflected Brownian motionon [0 , H (0)] , also with variance ( α + β ) λ per unit time and drift − γ . Inparticular, U ∗ = Λ H (0) ( W ∗ S ) = W ∗ S − K ∗ + + K ∗− , (6.22) where K ∗± are the unique RCLL nondecreasing functions satisfying K ∗± (0) =0 and Z [0 , ∞ ) I { U ∗ ( s ) Proof. Recall from Theorem 3.2 that c W ( n ) S ⇒ W ∗ S . Using the Sko-rokhod representation theorem, we construct the model primitives u ( n ) j , v ( n ) j and L ( n ) j for j ∈ N and n ∈ N on a common probability space (Ω , F , P ) suchthat the sequence of processes c W ( n ) S , n ∈ N , and the limiting process W ∗ S are defined on this space and (6.17) holds. Here and below the almost sureconvergences are in the J topology on D M [0 , ∞ ) or D R [0 , ∞ ), and since thelimits are continuous in every case, this is equivalent to uniform convergenceon compact intervals. Since the mapping f : D M [0 , ∞ ) D R [0 , ∞ ) given by f ( µ )( · ) = µ ( · )( R ) is continuous, we have (6.18). Under P the measure-valuedprocess W ∗ S constructed on Ω has the same distribution as the process W ∗ S appearing in Theorem 3.2, and thus W ∗ S takes values in the set of measure-valued process of the form R B ∩ [ F oS ( t ) , ∞ ) (1 − G ( y )) dy for some RCLL pro-cess F oS ( t ). However, W ∗ S ( t ) = R R ∩ [ F oS ( t ) , ∞ ) (1 − G ( y )) du = H ( F oS ( t )); hence KRUK, LEHOCZKY, RAMANAN AND SHREVE F oS ( t ) = F ∗ S ( t ) is given by (3.2). In other words, with F ∗ S defined by (3.2),the first equation in (3.3) holds. Due to Proposition 3.1, the above argumentalso shows that under P , W ∗ S is a Brownian motion with variance ( α + β ) λ per unit time and drift − γ . In addition, since for each t , the measure W ∗ S ( t )is nonatomic, we have (6.19).Now, following (4.1) and (6.3), we set U ( n ) = Φ( W ( n ) S ) and U ∗ = Φ( W ∗ S ).Also, as defined in (6.2), let b U ( n ) be the scaled version of U ( n ) , and let b U ( n ) and b U ∗ be as defined in the statement of the lemma. Then b U ( n ) , b U ( n ) , n ∈ N , U ∗ and U ∗ are also defined on (Ω , F , P ) and (6.20) follows from Lemma 4.1.This implies (6.21). Since U ∗ = Φ( W ∗ S )( R ) = Λ H (0) ( W ∗ S ), the characteriza-tion of U ∗ as a doubly reflected Brownian motion that satisfies relations(6.22) and (6.23) is a consequence of the statements following (6.4), in par-ticular, (6.5) and (6.6).Since the model primitives u ( n ) j , v ( n ) j and L ( n ) j for j ∈ N and n ∈ N are alldefined on (Ω , F , P ), so are the workload process W ( n ) and its scaled version c W ( n ) . Corollary 6.3 implies that b U ( n ) and c W ( n ) have the same limit, andhence (6.21), the almost sure counterpart to Theorem 3.4, holds. (cid:3) The assertion of Theorem 3.8 is that b R ( n ) W ⇒ K ∗ + , (6.24)where K ∗ + is the local time for U ∗ at H (0) from (6.22). For T < ∞ , define Z n ( T ) ∆ = { b R ( n ) U ( t ) = b R ( n ) W ( t ) , ≤ t ≤ T } . (6.25)From the workload evolution equations (4.69) and (4.70), it follows that if b U ( n ) ( t ) = c W ( n ) ( t ) for t ∈ [0 , T ], then b R ( n ) U ( t ) = b R ( n ) W ( t ) for t ∈ [0 , T ]. Hence,by Corollary 6.3, we know that for every T < ∞ , P ( Z n ( T )) → n → ∞ ,which shows that the limits in distribution of b R ( n ) U and b R ( n ) W , n ∈ N , mustcoincide (if they exist). Further, since b K ( n )+ = b R ( n ) U by Corollary 4.9, thesemust be equal to the limit in distribution of b K ( n )+ , n ∈ N . Hence, to completethe proof of Theorem 3.8, it suffices to show that b K ( n )+ ⇒ K ∗ + . (6.26)For n ∈ N and k ≥ 1, recall the definitions of τ ( n ) k − and σ ( n ) k given in (4.11)and (4.12), respectively, and define b τ ( n ) k − = n τ ( n ) k − and b σ ( n ) k ∆ = n σ ( n ) k . Applyingthe heavy traffic scaling to (4.13), it is easy to see that for t ≥ b K ( n )+ ( t ) = X k ∈ N hc W ( n ) S ( b σ ( n ) k − ) ∨ max s ∈ [ b σ ( n ) k ,t ∧ b τ ( n ) k ] c W ( n ) S ( s )( −∞ , − c W ( n ) S ( b σ ( n ) k − ) i . DF QUEUES WITH RENEGING Keeping in mind the limits in (6.17) and (6.19), we introduce the relatedprocess b Y ( n ) ( t ) ∆ = X k ∈ N h W ∗ S ( b σ ( n ) k ) ∨ max s ∈ [ b σ ( n ) k ,t ∧ b τ ( n ) k ] ( W ∗ S ( s ) − H (0)) + − W ∗ S ( b σ ( n ) k ) i (6.28)for t ≥ 0, and denote the difference by ε ( n ) ( t ) ∆ = b Y ( n ) ( t ) − b K ( n )+ ( t ) ∀ t ≥ . (6.29)Then b Y ( n ) is nondecreasing and continuous, and ε ( n ) is an RCLL process.In the next two lemmas, we show that b Y ( n ) increases only when U ∗ isat H (0) and that the difference ε ( n ) between b Y ( n ) and b K ( n )+ is negligible inheavy traffic. The main reason for introducing the sequence b Y ( n ) , n ∈ N , isthat it facilitates the proof of the former property. Lemma 6.7. For every n ∈ N , b Y ( n ) and b K ( n )+ are constant on each in-terval [ b τ ( n ) k − , b σ ( n ) k ) , k ≥ . Moreover, Z [0 ,T ] I { U ∗ ( t ) Fix n ∈ N . The first statement follows immediately from (6.27),(6.28), and the fact that the intervals [ b τ ( n ) k − , b σ ( n ) k ) and [ b σ ( n ) k , b τ ( n ) k ), k ≥ , ∞ ). Now, fix k ≥ J ( n ) k be the set ofpoints t ∈ [ b σ ( n ) k , b τ ( n ) k ) such that W ∗ S ( b σ ( n ) k ) ≤ max s ∈ [ b σ ( n ) k ,t ] ( W ∗ S ( s ) − H (0)) + = W ∗ S ( t ) − H (0) . (6.31)Since W ∗ S is continuous, J ( n ) k is closed, and so its complement in [ σ ( n ) k , τ ( n ) k )is the union of a countable number of open intervals, with possibly onehalf-open interval of the form [ b σ ( n ) k , a ) for some a > b σ ( n ) k . From the explicitformula for b Y ( n ) given in (6.28), it is easy to deduce that b Y ( n ) is also constanton each such interval. Thus, to establish (6.30), it only remains to show thatfor each k ≥ Z J ( n ) k I { U ∗ ( t ) Also, note thatsup s ∈ [0 , b σ ( n ) k ) h ( W ∗ S ( s ) − H (0)) + ∧ inf u ∈ [ s,t ] W ∗ S ( u ) i ≤ sup s ∈ [0 , b σ ( n ) k ) inf u ∈ [ s,t ] W ∗ S ( u ) ≤ W ∗ S ( b σ ( n ) k ) , and that the equality in (6.31) impliessup s ∈ [ b σ ( n ) k ,t ] h ( W ∗ S ( s ) − H (0)) + ∧ inf u ∈ [ s,t ] W ∗ S ( u ) i = W ∗ S ( t ) − H (0) . Since K ∗ ( t ) is equal to the maximum of the quantities on the left-hand sideof the last two displays, we conclude that K ∗ ( t ) ≤ W ∗ S ( b σ ( n ) k ) ∨ ( W ∗ S ( t ) − H (0)) = W ∗ S ( t ) − H (0) , where the equality follows from the inequality in (6.31). This, when combinedwith the fact that U ∗ ( t ) ∈ [0 , H (0)], shows that U ∗ ( t ) = W ∗ S ( t ) − K ∗ ( t ) = H (0) for all t ∈ J ( n ) k , which proves (6.32). (cid:3) We recall some standard definitions that will be used in the next lemma.Given f ∈ D [0 , ∞ ) and 0 ≤ t ≤ t < ∞ , the oscillation of f over [ t , t ] isOsc( f ; [ t , t ]) ∆ = sup {| f ( t ) − f ( s ) | : t ≤ s ≤ t ≤ t } , and the modulus of continuity of f over [0 , T ] is w f ( δ ; [0 , T ]) ∆ = sup {| f ( t ) − f ( s ) | : 0 ≤ s ≤ t ≤ T, | t − s | ≤ δ } . Lemma 6.8. As n → ∞ , ε ( n ) P −→ . Proof. Fix T > η > θ ( · ) satisfying lim δ ↓ θ ( δ ) = 0 andmajorizing the modulus of continuity w W ∗ ( · ; [0 , T ]) of the reflected Brownianmotion W ∗ over [0 , T ] on a set e Ω with P ( e Ω) ≥ − η .For each subsequence in N , there is a sub-subsequence S along which thelimits (6.17)–(6.21) hold P -almost surely. We choose e Ω so that these limitshold along S for all ω ∈ e Ω.In what follows, for n ∈ S , we denote Z n ( T ) simply by Z n , and evaluateall processes below at a fixed ω ∈ Z n ∩ e Ω. Choose ∆ < y ∗ / 3, and let n ∈ S DF QUEUES WITH RENEGING be such that for all n ∈ S , n ≥ n ,sup t ∈ [0 ,T ] | c W ( n ) S ( t )( −∞ , − ( W ∗ S ( t ) − H (0)) + | ≤ ∆ , (6.34) sup t ∈ [0 ,T ] | c W ( n ) S ( t − ) − W ∗ S ( t ) | ≤ ∆ , (6.35) sup t ∈ [0 ,T ] | c W ( n ) ( t − ) − W ∗ ( t ) | ≤ ∆ , sup t ∈ [0 ,T ] | b U ( n ) ( t − ) − U ∗ ( t ) | ≤ ∆ . (6.36)From the definitions (6.27) and (6.28), respectively, of b K ( n )+ and b Y ( n ) it isclear that, for every k ∈ N such that τ ( n ) k ≤ T ,sup t ∈ [ b σ ( n ) k , b τ ( n ) k ] | b Y ( n ) ( t ) − b Y ( n ) ( b σ ( n ) k − ) − ( b K ( n )+ ( t ) − b K ( n )+ ( b σ ( n ) k − )) | ≤ . Define J n ∆ = { k ∈ N : b K ( n )+ ( b τ ( n ) k ) − b K ( n )+ ( b σ ( n ) k − ) > , b τ ( n ) k ≤ T } , e J n ∆ = { k ∈ N : b Y ( n ) ( b τ ( n ) k ) − b Y ( n ) ( b σ ( n ) k − ) > , b τ ( n ) k ≤ T } , and let c ( n ) be the cardinality of J ( n ) ∪ e J ( n ) . Since b K ( n )+ and b Y ( n ) are bothconstant on intervals of the form [ b τ ( n ) k − , b σ ( n ) k ), k ≥ ε ( n ) ( T ) ∆ = sup s ∈ [0 ,T ] | b Y ( n ) ( s ) − b K ( n )+ ( s ) | ≤ c ( n ) ∆ . (6.37)We now claim that k ∈ [ J n ∪ e J n ] ⇒ Osc( W ∗ , [ b σ ( n ) k , b τ ( n ) k ]) ≥ y ∗ . (6.38)We defer the proof of the claim and instead first show that the lemma fol-lows from this claim. Let θ − ( · ) denote the inverse of θ and define M ∆ = T /θ − ( y ∗ / < ∞ . From the claim, we conclude that if k ∈ [ J n ∪ e J n ] then b τ ( n ) k − b σ ( n ) k ≥ θ − ( y ∗ / > 0, which in turn implies that c ( n ) ≤ M . Substitut-ing this into (6.37), we conclude that for every ∆ > 0, there exists n (∆) ∈ S such that for all n ∈ S , n ≥ n (∆), P ( ε ( n ) ( T ) > M ∆) ≤ P ( Z cn ∪ e Ω c ) ≤ P ( Z cn ) + η. Taking limits as n → ∞ through S and using the fact that P ( Z n ) → ε ( n ) ( T ) P −→ 0. We have shown that for each subsequence KRUK, LEHOCZKY, RAMANAN AND SHREVE in N , there is a sub-subsequence along which ε ( n ) ( T ) P −→ 0. It follow that ε ( n ) ( T ) P −→ 0, where the limit is taken over all n ∈ N , and this proves thelemma.We now turn to the proof of the claim (6.38). Note first that by the defi-nition of H (0) and y ∗ , we have H (0) ≥ y ∗ . If k ∈ e J n , then Lemma 6.7 showsthat U ∗ ( t ) = H (0) for some t ∈ [ b σ ( n ) k , b τ ( n ) k ). By the equality b U ( n ) ( b σ ( n ) k − ) = 0proved in Lemma 4.7 and (6.36), this implies that the oscillation of U ∗ on[ b σ ( n ) k , b τ ( n ) k ) is no less than H (0) − ∆ ≥ y ∗ / 3. Since W ∗ = U ∗ , the conclusionin (6.38) holds.Finally, suppose k ∈ J n . Since b K ( n )+ = b R ( n ) U = b R ( n ) W , we have b R ( n ) W ( b τ ( n ) k ) − b R ( n ) W ( b σ ( n ) k − ) > , that is, the deadline of a customer in the reneging system expires during theunscaled time interval [ σ ( n ) k , τ ( n ) k ]. Since W ( n ) ( σ ( n ) k − ) = 0(6.39)[because U ( n ) ( σ ( n ) k − ) = 0 and, by Lemma 5.2, W ( n ) ≤ U ( n ) ], this customermust arrive during the interval [ σ ( n ) k , τ ( n ) k ). Since his initial lead time isgreater than or equal to √ ny ∗ , there is a time nt ∈ [ σ ( n ) k , τ ( n ) k ) when thiscustomer has lead time exactly √ ny ∗ . After time nt , this customer cannotbe preempted by new arrivals, all of which have initial lead times greaterthan or equal to √ ny ∗ . At time nt , the work that must be completed beforethis customer is served to completion is at most W ( n ) ( nt )(0 , √ ny ∗ ]. Sincethis customer becomes late, we must have W ( nt ) ≥ W ( n ) ( nt )(0 , √ ny ∗ ] > √ ny ∗ , or equivalently, c W ( n ) ( t ) ≥ c W ( n ) ( t )(0 , y ∗ ] > y ∗ . By right continuity, c W ( n ) (( t + ν ) − ) > y ∗ for some ν > t + ν ≤ b τ ( n ) k . From thesecond inequality in (6.35) and the fact that c W ( n ) ( b σ ( n ) k − ) = 0 [the scaledversion of (6.39)], we conclude that W ∗ ( t + ν ) − W ∗ ( b σ ( n ) k ) ≥ y ∗ , and this gives us the conclusion in (6.38). (cid:3) Proof of Theorem 3.8. Fix T < ∞ . Let δ ( n ) ∆ = U ∗ − b U ( n ) , and let δ ( n ) ∆ = sup s ∈ [0 ,T ] | U ∗ ( s ) − b U ( n ) ( s ) | . According to (4.6) and (4.15), U ( n ) = W ( n ) S − K ( n )+ + K ( n ) − . We scale this equation to obtain U ∗ = c W ( n ) S + δ ( n ) − b K ( n )+ + b K ( n ) − = c W ( n ) S + δ ( n ) + ε ( n ) − b Y ( n ) + b K ( n ) − , (6.40) DF QUEUES WITH RENEGING where [cf. (4.14)] b K ( n ) − ( t ) ∆ = − X k ∈ N [( c W ( n ) S ( b τ ( n ) k − ) − ( b σ ( n ) k ∧ t − b τ ( n ) k − )) + − c W ( n ) S ( b τ ( n ) k − )] , b K ( n )+ is defined by (6.27) and ε ( n ) is defined by (6.29). According to (4.16), R T I { b U ( n ) ( t ) > } d b K ( n ) − ( t ) = 0, which implies Z T I { U ∗ ( t ) >δ ( n ) } d b K ( n ) − ( t ) = 0 . (6.41)Since c W ( n ) S + δ ( n ) + ǫ ( n ) ⇒ W ∗ S due to (6.18), (6.20) and Lemma 6.8, and,by (6.22), U ∗ is obtained by applying the Skorokhod map on [0 , H (0)] to W ∗ S , the convergence (6.26) is an immediate consequence of (6.40), (6.41),Lemmas 6.7, 6.8 and the invariance principle for reflected Brownian motions.However, since we are in a particularly simple setting here, we will providea direct proof without invoking the general invariance principle.We choose n so that δ ( n ) < H (0) / ρ = 0, and for k ≥ ν k = min (cid:26) t ≥ ρ k − (cid:12)(cid:12)(cid:12) U ∗ ( t ) = 2 H (0)3 (cid:27) , ρ k = min (cid:26) t ≥ ν k (cid:12)(cid:12)(cid:12) U ∗ ( t ) = H (0)3 (cid:27) . Then 0 = ρ < ν < ρ < ν < · · · and lim k →∞ ρ k = lim k →∞ ν k = ∞ . For n ≥ n , b K ( n ) − is constant on each of the intervals [ ν k , ρ k ]. Moreover, Lemma 6.7implies that for each k , b Y ( n ) is constant on each of the intervals [ ρ k − , ν k ].For t ∈ [ ν k , ρ k ], we have from (6.40), (6.18), (6.20) and Lemma 6.8 that b Y ( n ) ( t ) − b Y ( n ) ( ν k ) = c W ( n ) S ( t ) − U ∗ ( t ) + δ ( n ) ( t ) + ε ( n ) ( t ) − c W ( n ) S ( ν k ) + U ∗ ( ν k ) − δ ( n ) ( ν k ) − ε ( n ) ( ν k ) P −→ W ∗ S ( t ) − U ∗ ( t ) − ( W ∗ S ( ν k ) − U ∗ ( ν k )) . It follows that, uniformly for t ∈ [0 , T ], b Y ( n ) ( t ) converges in probability to X k ∈ N [ W ∗ S (( t ∨ ν k ) ∧ ρ k ) − U ∗ (( t ∨ ν k ) ∧ ρ k ) − ( W ∗ S ( ν k ) − U ∗ ( ν k ))] . (6.42)However, (6.23) implies that for each k , K ∗− is constant on [ ν k , ρ k ], and K ∗ + is constant on [ ρ k − , ν k ]. Therefore, (6.22) implies that for t ∈ [ ν k , ρ k ], K ∗ + ( t ) − K ∗ + ( ν k ) = W ∗ S ( t ) − U ∗ ( t ) − ( W ∗ S ( ν k ) − U ∗ ( ν k )) . This implies that the expression in (6.42) is K ∗ + ( t ). But b Y ( n ) and b K ( n )+ havethe same limit in probability because of Lemma 6.8, and we conclude thatmax t ∈ [0 ,T ] | b K ( n )+ ( t ) − K ∗ + ( t ) | P −→ . (6.43) KRUK, LEHOCZKY, RAMANAN AND SHREVE Convergence in probability implies weak convergence, and we have (6.26). (cid:3) 7. Performance evaluation and simulations. We use the heavy trafficapproximations of this paper to evaluate the performance of the system withreneging and compare this to the system in which all customers are servedto completion. The predictions of the theory, derived in Section 7.1 andcompared to simulations in Section 7.2, are predicated on the assumptionthat one can interchange the limit as n → ∞ and the limit as time goesto infinity of the fraction of reneged work. A formal proof would require acoupling argument such as that found in [37]. The simulation results attestto the accuracy of the approximations derived in Section 7.1 and also showthe great difference in performance between the reneging and nonrenegingsystems.7.1. Derivation of theory predictions. We derive formulas (1.1)–(1.7). Webegin with one of the main results of this paper, Theorem 3.4, which statesthat the limiting scaled workload in the reneging system is a reflected Brow-nian motion in [0 , H (0)] with drift. More specifically, W ∗ ( t ) = W ∗ S ( t ) − K ∗ + ( t ) + K ∗− ( t ) , (7.1)where W ∗ S ( t ) is a reflected Brownian motion on [0 , ∞ ) with variance σ = λ ( α + β ) per unit time and drift − γ , K ∗− is the nondecreasing processstarting at K ∗− (0) = 0 that grows only when W ∗ = 0, and K ∗ + is the nonde-creasing process starting at K ∗ + (0) = 0 that grows only when W ∗ = H (0).We further saw in Theorem 3.8 that K ∗ + ( t ) is the limit of the scaled work-load that reneges prior to time t in the diffusion scaling, that is, √ nK ∗ + ( t ) isapproximately the (unscaled) workload that reneges in the n th system priorto time nt . Lemma 7.1 ([15], Proposition 5, page 90). We have lim t →∞ t K ∗ + ( t ) = lim t →∞ t E K ∗ + ( t ) = γe γH (0) /σ − , if γ = 0 , σ H (0) , if γ = 0 . (7.2) Proof. The first equality in (7.2) is a consequence of the fact that W ∗ has a stationary distribution [see (7.5) below]. For the proof of thesecond equality, recall that W ∗ S has the decomposition (2.16). Let f be a C function. Applying Itˆo’s formula to f ( W ∗ ( t )) and taking expectations, we DF QUEUES WITH RENEGING obtain f ′ (0) E [ I ∗ S ( t ) + K ∗− ( t )] − f ′ ( H (0)) E K ∗ + ( t )= E Z t (cid:20) γf ′ ( W ∗ ( s )) − σ f ′′ ( W ∗ ( s )) (cid:21) ds (7.3) + E f ( W ∗ ( t )) − f (0) . Taking f ( x ) = x , we obtain E [ I ∗ S ( t ) + K ∗− ( t )] − E K ∗ + ( t ) = γt + E W ∗ ( t ) − f (0).If γ = 0, we may take f ( x ) = σ γ e γx/σ in (7.3), which leads to the equation E [ I ∗ S ( t ) + K ∗− ( t )] − e γH (0) /σ E K ∗ + ( t ) = σ γ ( E e γW ∗ ( t ) /σ − E K ∗ + ( t ), we obtain the second equality in (7.2) for γ = 0. Toobtain this equality for γ = 0, we take f ( x ) = x . (cid:3) According to (2.12), the work that arrives to the n th system by time nt is V ( n ) ( A ( n ) ( nt )) = √ n b N ( n ) ( t ) + nt . But, b N ( n ) is approximately N ∗ , and hencelim t →∞ √ n b N ( n ) ( t ) + ntnt ≈ lim t →∞ √ nN ∗ ( t ) + ntnt = (cid:18) − γ √ n (cid:19) . Therefore, if γ = 0, the long-run fraction of reneged work is approximatelylim t →∞ √ nK ∗ + ( t ) V ( n ) ( A ( n ) ( nt )) = 1 √ n lim t →∞ t K ∗ + ( t ) · lim t →∞ (cid:18) √ n b N ( n ) ( t ) + ntnt (cid:19) − ≈ γ/ √ n (1 − γ/ √ n )( e γH (0) /σ − . Finally, (2.4) implies that the expected lead time in the n th system is E L ( n ) j = R ∞ (1 − G ( y/ √ n )) dy = √ nH (0). Using this formula and (2.10), weconclude that the fraction of work that reneges in the n th system when γ = 0is approximately 1 − ρ ( n ) ρ ( n ) ( e − ρ ( n ) ) E L ( n ) j /σ − 1) = 1 − ρ ( n ) ρ ( n ) ( e θD − , (7.4)where θ = 2(1 − ρ ( n ) ) σ ≈ γ √ nσ , D = E L ( n ) j = √ nH (0) . We have suppressed the dependence of θ and D on n , which will remainfixed. If γ = 0, then in place of (7.4) we have σ D . We have established (1.1)and (1.2). KRUK, LEHOCZKY, RAMANAN AND SHREVE Remark 7.2. Corollary 3.7 also implies that the limiting scaled queuelength process is λW ∗ , which is a doubly reflected Brownian motion in[0 , λH (0)] with drift − γλ and variance per unit time λ σ . This incorrectlysuggests that λ √ nK ∗ + ( t ) is approximately the number of customers whorenege in the n th system prior to nt . The simulations indicate that thisnaive interpretation of Corollary 3.7 applied to the queue length process isincorrect, as does the following heuristic.According to [15], Proposition 5, page 90, if γ = 0, the stationary densityfor W ∗ is ϕ ∗ ( x ) ∆ = γe − γx/σ σ (1 − e − γH (0) /σ ) , if 0 ≤ x ≤ H (0),0 , otherwise,(7.5)whereas the stationary density is uniform on [0 , H (0)] if γ = 0. Therefore,for γ = 0 and t large, the density of W ( n ) ( nt ) ≈ √ nW ∗ ( t ) is approximately ϕ ( w ) = 1 √ n ϕ ∗ ( w/ √ n ) = θe − θw − e − θD , if 0 ≤ w ≤ D ,0 , otherwise.We have suppressed the dependence of ϕ on n .Suppose now that the lead times of arriving customers are not random.Then in the n th system, all lead times are equal to √ nH (0) = D . In thiscase, the EDF policy serves customers in order of arrival (FIFO). Supposethe workload in queue is W at the time of arrival of a customer whoseservice requirement is V . Recall that the expected service time is 1 /µ ( n ) ,and because n is fixed, we suppress it and write E V = 1 /µ . The arrivingcustomer will be served to completion if and only if W + V ≤ D . Supposefurther that the arrival process A ( n ) is Poisson, so that according to thePASTA property (“Poisson arrivals see time averages”; see [1], Theorem6.7, page 218), an arriving customer will encounter a workload W havingapproximately the distribution ϕ . The probability the arriving customereventually reneges is thus P { W > D − V } = E [ P { W > D − V | V } ] = E (cid:20)Z D ( D − V ) + ϕ ( w ) dw (cid:21) . Because D is of order √ n and V is of order 1, we have ( D − V ) + = D − V with high probability. Using this approximation, we complete the calculationfor the case γ = 0 to obtain P { Customer reneges } ≈ e θD − E e θV − . (7.6) DF QUEUES WITH RENEGING If the customer reneges, then work V + W − D > E [ V + W − D | Customer reneges] ≈ E (cid:20) R D ( D − V ) + ( V + w − D ) ϕ ( w ) dw P { Customer reneges } (cid:21) . Again using the approximation ( D − V ) + ≈ D − V , we obtain E [ V + W − D | Customer reneges] ≈ θ − E V E e θV − ≈ θ − E Vθ E V + 1 / θ E [ V ] + O ( n − / ) ≈ E [ V ]2 E V . The last expression is, perhaps not surprisingly, the formula for the averageresidual lifetime of a renewal cycle (see [32], Example 3.6(b), pages 80 and81). Consequently, when lead times are constant and the arrival process isPoisson, we should expect the total number of customers reneging in [0 , t ]times the expected amount of work lost per reneging customer to approxi-mately equal the total amount of work lost by reneging in [0 , t ]. If we divideboth by the total number of customer arrivals in [0 , t ] and take limits as t → ∞ , we findFraction of lost customers in reneging system ≈ Fraction of lost work in reneging system E [ V + W − D | Customer reneges] ∗ E V (7.7) ≈ E V ) E [ V ] × (Fraction of lost work in reneging system) . This is (1.7) with E V = µ and E [ V ] = β + µ .If V is exponentially distributed, hence E [ V ] = 2( E V ) , then (7.7) impliesthat the fraction of customers who renege will be approximately the fractionof work that reneges. See Figure 1 for simulations that confirm this assertion.On the other hand, if V is nonrandom, hence equal to its mean 1 /µ , then(7.7) predicts that the fraction of customers who renege will be twice thefraction of work that reneges. See Figure 2 for simulations that confirm thisassertion. Both these conclusions hold irrespective of the value of λ .The last conclusion is inconsistent with a naive interpretation of Corollary3.7, according to which work reneges at a rate 1 /λ times the rate of customerreneging. Since work arrives at a rate E V ≈ /λ times the rate of customerarrivals, this naive interpretation of Corollary 3.7 would say that the fractionof work reneging would approximately agree with the fraction of customersreneging regardless of the distribution of V . KRUK, LEHOCZKY, RAMANAN AND SHREVE Fig. 1. M/M/ queue. Fig. 2. M/D/ queue. DF QUEUES WITH RENEGING We next turn our attention to the performance of the standard (nonreneg-ing) system. Recall from (2.15) that the scaled workload process when allcustomers are served to completion converges to W ∗ S , a reflected Brownianmotion with drift − γ (we now assume γ > σ . In particular, W ( n ) S ( nt ) ≈ √ nW ∗ S ( t ). Thestationary density for W ∗ S is ϕ ∗ S ( x ) ∆ = ( γσ e − γx/σ , if x ≥ , otherwise,and so for large t , the density of W ( n ) ( nt ) is approximately ϕ S ( w ) = 1 √ n ϕ ∗ S ( w/ √ n ) = (cid:26) θe − θw , if w ≥ , otherwise.Consequently, the long-run fraction of time W ( n ) spends above level D is e − θD . The workload level at which the limiting frontier reaches 0 is H (0),and hence it is approximately the case that the n th system sees lateness ifand only if W ( n ) exceeds D = √ nH (0). In other words, the theory predictsthat Fraction of late customers in standard system= Fraction of late work in standard system(7.8) = e − θD . We are using here the result for GI/G/ n →∞ lim T →∞ T Z T I { c W ( n ) S ( t ) >H (0) } dt = lim T →∞ lim n →∞ T Z T I { c W ( n ) S ( t ) >H (0) } dt = lim T →∞ T Z T I { W ∗ S ( t ) >H (0) } dt, a result that grows out of the work of Kingman [19, 20] (see [10] for a generalresult that specializes to the case under consideration).It is important to compare the fraction of work that reneges in the reneg-ing system, given by (7.4), with the fraction of work that is late in thestandard (nonreneging) system. The ratio of these quantities of lost/latework is Lost work in reneging systemLate work in standard system ≈ e θD e θD − (cid:18) − ρ ( n ) ρ ( n ) (cid:19) . (7.9) KRUK, LEHOCZKY, RAMANAN AND SHREVE The parameter θ is O (1 / √ n ), θD is O (1), and 1 − ρ ( n ) is O (1 / √ n ). Thusthe ratio in (7.9) is O (1 / √ n ). Remark 7.3. If lead times are a nonrandom constant D , EDF reducesto first-in-first-out, and the fraction of lost customers in an M/G/1 queuewith 0 < ρ < − ρ ) P { W > D } / (1 − ρ P { W > D } ), where W is thesteady-state workload in the corresponding nonreneging M/G/1 queue (see[3]). In the heavy traffic limit of our model, P { W > D } = e − θD [see thederivation of (7.8)]. Recalling that 1 − ρ = O (1 / √ n ) in (1.1), we observethat this is consistent with (1.1).7.2. Simulation results. We conducted a simulation study to assess theaccuracy of these approximations and to compare the performance of the sys-tems with and without reneging. Two systems were considered, an M/M/ λ = 0 . µ = 1 . 96, and so the traffic intensity is ρ = 0 . θ = 0 . M/M/ θ = 0 . 02 for the M/D/1 case. The initial deadline distribution is uniform on[5 , B ] with the mean deadline D = B , varying from B = 5 (constant dead-lines) to B = 200. The data points are the simulation results averaged overone billion customer arrivals per case. The curves that are superimposed onthe data are the theoretical values, e − θD for the case in which customers areserved to completion (the standard system), and equations (1.1) and (1.7)for the fraction of work lost and the fraction of lost customers for the reneg-ing system. Equation (1.7) is derived in Remark 7.2 under the assumptionof constant deadlines. Nevertheless, we apply it for the variable deadlinecase in the simulation study. The fraction of late work or late customers forthe system in which customers are served to completion is also presented tocompare its performance with that of the reneging system.The M/M/ y -axis against themean deadline on the x -axis. There is nearly perfect agreement betweenthe theoretical approximation and the simulation. In fact, one cannot seethe plot of “Fraction of Customers Late (No Reneging)” because it coin-cides with the “Theory” plot at the top of the figure. Similarly, one can seeonly parts of the plots of “Fraction of Customers Reneging” and “Fractionof Work Reneging” because they coincide with the “Theory” plot in themiddle of the figure. One can see the linear form for the case of service tocompletion. Furthermore, the simulation confirms the prediction of (1.1)–(1.4) that for sufficiently large values of D , the performance of the renegingsystem is parallel on a log scale to that of the standard system with the two DF QUEUES WITH RENEGING curves separated by approximately 0 . 02. This corresponds to a reduction inwork that misses its deadline by a factor of 40 to 50.Figure 2 presents the results for the M/D/1 system. The results are qual-itatively identical to those of Figure 1, except the fits of the theoreticalcurves are not as exact as the fits for the M/M/ θ = 0 . 02 is slightly too small and hence the theory slightlyoverestimates the fraction of work that misses its deadline, especially whenthe mean deadline is large. Also, the lost or late work and the customer lossor lateness fractions are significantly smaller than for the M/M/ D is again a factor of 40 to 50. In both figures, it is clear thatthere are one to two orders of magnitude of improvement in the overall per-formance of the system resulting from stopping service on customers whentheir deadlines expire.APPENDIX: OPTIMALITY OF EDF Proof of Theorem 5.1. Let π be a service policy and let t be thefirst time π deviates from the EDF policy, either because it idles when thereis work present, or it serves a customer other than the customer present withthe smallest lead time. Let j be the index of the customer with the smallestlead time at time t .We consider first the case that π idles at time t . In this case, we define ρ ( π ) to be the policy that emulates π except as noted below. From time t , whenever π idles, ρ ( π ) serves customer j , at least until time t , whencustomer j leaves the ρ ( π ) system because either ρ ( π ) serves customer j tocompletion or else the deadline of customer j elapses. From time t , ρ ( π )idles if π serves customer j . We will show that for t ≥ R ρ ( π ) ( t ) ≤ R π ( t ) . (A.1)Let v k ( t ) [resp., v ρk ( t )] be the residual service time of the k th customer attime t under π [resp., ρ ( π )]. In particular, if d k is the deadline of the k thcustomer, then v k ( d k − ) [resp., v ρk ( d k − )] is the work corresponding to thiscustomer that is deleted by π [resp., ρ ( π )] due to lateness, and R ρ ( π ) ( t ) = X k : d k ≤ t v ρk ( d k − ) , R π ( t ) = X k : d k ≤ t v k ( d k − ) . (A.2)By the definition of ρ ( π ), for t ≥ k = j , we have v ρk ( t ) = v k ( t ) , (A.3)whereas v ρj ( t ) ≤ v j ( t ) . (A.4) KRUK, LEHOCZKY, RAMANAN AND SHREVE Summing (A.3) over k = j , invoking (A.4) and (A.2), we obtain (A.1).We next consider the case that at time t , π serves customer i = j . Inthis case, we define ρ ( π ) to be the policy that emulates π except as notedbelow. From time t , whenever π serves customer i , ρ ( π ) serves customer j , at least until time t , when ρ ( π ) serves customer j to completion or thedeadline of customer j elapses. From time t , ρ ( π ) serves customer i if π serves customer j , provided customer i is present in the system under ρ ( π ). If π serves customer j and customer i is not present under ρ ( π ), then ρ ( π ) idles.We again have (A.2) and (A.4), whereas (A.3) now holds only for k / ∈ { i, j } .If the i th customer is served to completion under ρ ( π ), then v ρi ( d i − ) = 0,and (A.3) for k / ∈ { i, j } , and (A.4) imply that (A.1) holds for all t . It remainsto consider the case that the i th customer becomes late under ρ ( π ). In thiscase (A.3) for k / ∈ { i, j } and (A.4) imply that (A.1) holds for t ∈ [0 , d i ). Let w denote the work done by ρ ( π ) on the j th customer when π works on the i th customer in the interval [ t , t ). Let w be the work done by ρ ( π ) oncustomer i in the time interval [ t , ∞ ) while π works on customer j in thistime interval. Finally, let w be the work done by π on customer j in the timeinterval [ t , ∞ ) while ρ ( π ) is idle. Then v ρj ( d j − ) + w = v j ( d j − ) + w + w and v ρi ( d i − ) + w = v i ( d i − ) + w , which implies v ρj ( d j − ) + v ρi ( d i − ) = v j ( d j − ) + v i ( d i − ) + w . (A.5)We argue by contradiction that w cannot be positive. If w were positive,then at some time t ≥ t , π serves customer j and customer i is not in the ρ ( π ) system. This implies that d j > t , and since by assumption, d i > d j , theabsence of customer i in the ρ ( π ) system means that this system has servedcustomer i to completion. We conclude that v ρi ( d i − ) = 0. On the other hand,customer j is also not in the ρ ( π ) system at time t ≥ t , and so v ρj ( d j − ) = 0as well. The left-hand side of (A.5) is zero, and hence w must be zero. Weconclude that v ρj ( d j − ) + v ρi ( d i − ) = v j ( d j − ) + v i ( d i − ) . (A.6)Since d j < d i , if t ≥ d i , then (A.3) for k / ∈ { i, j } and (A.6) imply (A.1).Starting from the service policy π , we have obtained a service policy ρ ( π )that either is work conserving until the departure of customer j or elsegives customer j priority over customer i until the departure of customer j .However, immediately after time t , the policy π may serve some customer k / ∈ { i, j } , and hence ρ ( π ) also serves k at this time, although customer j is more urgent. Therefore, we apply n iterations of the mapping ρ , where n is the number of customers in the π system at time t , and thereby obtaina policy that is work-conserving and serves in EDF order at least until thefirst time after t that there is a departure or an arrival. We have R ρ n ( π ) ( t ) ≤ R π ( t ) for all t ≥ DF QUEUES WITH RENEGING By assumption, for each t the number of system arrivals A ( t ) by time t is finite. Hence the maximum number of customers in the system over theinterval [0 , t ] is bounded by A ( t ), and the number of arrivals and departuresup to time t is bounded by 2 A ( t ), irrespective of the service policy. Thus,if we start with any policy π , the number of iterations of the mapping ρ required to obtain a policy that is work conserving and serves in EDF orderup to time t is finite. Under this policy the amount of work removed bylateness up to time t is the same as for the EDF system in the theorem, andhence (5.1) holds. (cid:3) Remark A.1. In the above proof we have implicitly assumed that π [and thus ρ ( π )] never serves more than one customer at the same time. Thisassumption simplifies the exposition of the argument, and the generality ofTheorem 5.1 is sufficient for this paper. However, the proof can be general-ized to policies permitting simultaneous service of customers (e.g., processorsharing). In this case, in the construction of ρ ( π ) we must additionally takethe rates at which customers receive service into account. For example, thedifference in the rates with which the j th customer receives service under ρ ( π ) and π in the time interval [ t , t ) must be equal to the rate of serviceof the i th customer under π in this time interval, the rates of service of allother customers in this time interval under π and ρ ( π ) must be the same,etc. REFERENCES [1] Asmussen, S. (2003). Applied Probability and Queues , 2nd ed. Applications of Math-ematics (New York) . Springer, New York. MR1978607[2] Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley, NewYork. MR1700749[3] Boots, N. and Tijms, H. (1999). A multiserver queueing system with impatientcustomers. Management Science Chen, H. and Mandelbaum, A. (1991). Stochastic discrete flow networks: Diffusionapproximations and bottlenecks. Ann. Probab. Decreusefond, L. and Moyal, P. (2008). Fluid limit of a heavily loaded EDFqueue with impatient customers. Markov Process. Related Fields Down, D. , Gromoll, H. C. and Puha, A. (2009). Fluid limits for shortest remainingprocessing time queues. Math. Operations Research Doytchinov, B. , Lehoczky, J. and Shreve, S. (2001). Real-time queues in heavytraffic with earliest-deadline-first queue discipline. Ann. Appl. Probab. Dupuis, P. and Ramanan, K. (1999). Convex duality and the Skorokhod problem. Probab. Theory Related Fields Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization andConvergence . Wiley, New York. MR838085 KRUK, LEHOCZKY, RAMANAN AND SHREVE[10] Gamarnik, D. and Zeevi, A. (2006). Validity of heavy traffic steady-state ap-proximation in generalized Jackson networks. Ann. Appl. Probab. Gromoll, H. C. (2004). Diffusion approximation for a processor sharing queue inheavy traffic. Ann. Appl. Probab. Gromoll, H. C. and Kruk, L. (2007). Heavy traffic limit for a processor sharingqueue with soft deadlines. Ann. Appl. Probab. Gromoll, H. C. , Kruk, L. and Puha, A. Diffusion limits for shortest remainingprocessing time queues. Preprint, Dept. Mathematics, Univ. Virginia. Availableat http://arxiv.org/abs/1005.1035 .[14] Harrison, J. M. and Reiman, M. Reflected Brownian motion in an orthant. Ann.Probab. Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems . Wiley, NewYork. MR798279[16] Iglehart, D. L. and Whitt, W. (1970). Multiple channel queues in heavy traffic.I. Adv. in Appl. Probab. Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus . Graduate Texts in Mathematics . Springer, New York. MR917065[18] Kaspi, H. and Ramanan, K. (2011). Law of large numbers limits for many-serverqueues. Ann. Appl. Probab. Kingman, J. F. C. (1961). The single server queue in heavy traffic. Proc. CambridgePhilos. Soc. Kingman, J. F. C. (1962). On queues in heavy traffic. J. Roy. Statist. Soc. Ser. B Kruk, L. (2007). Diffusion approximation for a G/G/ Ann. Univ. Mariae Curie-Sk lodowska Math. A Kruk, L. , Lehoczky, J. P. and Shreve, S. (2003). Second order approximationfor the customer time in queue distribution under the FIFO service discipline. Ann. Univ. Mariae Curie-Sk lodowska Sect. AI Inform. Kruk, L. , Lehoczky, J. P. , Shreve, S. and Yeung, S.-N. (2004). Earliest-deadline-first service in heavy-traffic acyclic networks. Ann. Appl. Probab. Kruk, L. , Lehoczky, J. P. and Shreve, S. (2006). Accuracy of state space collapsefor earliest-deadline-first queues. Ann. Appl. Probab. Kruk, L. , Lehoczky, J. P. , Ramanan, K. and Shreve, S. E. (2007). An explicitformula for the Skorokhod map on [0 , a ]. Ann. Probab. Kruk, L. , Lehoczky, J. P. , Ramanan, K. and Shreve, S. (2008). Double Sko-rokhod map and reneging real-time queues. In Markov Processes and RelatedTopics: A Festschrift for Thomas G. Kurtz . Inst. Math. Stat. Collect. Lehoczky, J. P. (1996). Real-time queueing theory. In Proc. of IEEE Real-TimeSystems Symposium Limic, V. (2000). On the behavior of LIFO preemptive resume queues in heavy traffic. Electron. Comm. Probab. Panwar, S. S. and Towsley, D. (1992). Optimality of the stochastic earliest dead-line policy for the G/M/c queue serving customers with deadlines. In SecondORSA Telecommunications Conference . ORSA (Operations Research Society ofAmerica), Baltimore, MD.[30] Prokhorov, Y. (1956). Convergence of random processes and limit theorems inprobability theory. Theory Probab. Appl. [31] Ramanan, K. and Reiman, M. I. (2003). Fluid and heavy traffic diffusion lim-its for a generalized processor sharing model. Ann. Appl. Probab. Ross, S. M. (1983). Stochastic Processes . Wiley, New York. MR683455[33] Ward, A. R. and Glynn, P. W. (2003). A diffusion approximation for a Markovianqueue with reneging. Queueing Syst. Ward, A. R. and Glynn, P. W. (2005). A diffusion approximation for a GI/GI/ Queueing Syst. Whitt, W. (1971). Weak convergence theorems for priority queues: Preemptive-resume discipline. J. Appl. Probab. Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-ProcessLimits and Their Application to Queues . Springer, New York. MR1876437[37] Zhang, J. and Zwart, B. (2008). Steady state approximations of limited processorsharing queues in heavy traffic. Queueing Syst. L. KrukDepartment of MathematicsMaria Curie-Sklodowska UniversityLublinPolandandInstitute of MathematicsPolish Academy of SciencesWarsawPolandE-mail: [email protected] J. LehoczkyDepartment of StatisticsCarnegie Mellon UniversityPittsburgh, Pennsylvania 15213USAE-mail: [email protected] K. RamananDivision of Applied MathematicsBrown UniversityProvidence, Rhode Island 02912USAE-mail: Kavita [email protected]