An Infinite Dimensional Model for a Many Server Priority Queue
AAn Infinite Dimensional Model forA Many Server Priority Queue
Neal Master, Zhengyuan Zhou, and Nicholas Bambos
Department of Electrical Engineering, Stanford UniversityStanford, CA 94305 { nmaster, zyzhou, bambos } @stanford.edu Abstract —We consider a Markovian many server queueingsystem in which customers are preemptively scheduled accordingto exogenously assigned priority levels. The priority levels arerandomly assigned from a continuous probability measure ratherthan a discrete one and hence, the queue is modeled by aninfinite dimensional stochastic process. We analyze the equilib-rium behavior of the system and provide several results. Wederive the Radon-Nikodym derivative (with respect to Lebesguemeasure) of the measure that describes the average distributionof customer priority levels in the system; we provide a formulafor the expected sojourn time of a customer as a function ofhis priority level; and we provide a formula for the expectedwaiting time of a customer as a function of his priority level. Weverify our theoretical analysis with discrete-event simulations.We discuss how each of our results generalizes previous work oninfinite dimensional models for single server priority queues.
I. I
NTRODUCTION
Priority queueing models arise in several applications. Inpacket switched communication networks, priority levels areused to deliver differentiated levels of quality of service, e.g.[1], [2]. In emergency medicine, priority queueing modelsare used to study triage policies, e.g. [3]. Priority queueingmodels are also used in financial engineering to model orderbooks in which limit orders are given priority for matchingwith other orders based on their price and time of arrival[4]. Because priority queueing models are useful is so manydomains, several priority queueing models exist; see [5] for astandard reference on stochastic priority queueing.In this paper, we formulate and analyze an
M/M/c priorityqueueing model in which priority levels are drawn from a continuum . Unlikely previous models that allow for finitelymany priority levels, e.g. [6], [7], our model requires an infinitedimensional state process. Consequently, standard Markovchain techiques that apply to finitely many priority levels, e.g.[8], do not apply. Our recent previous work [9] also considereda continuous distribution of priority levels but only for thesingle server case. The current paper generalizes the resultsin our previous work [9] by extending the results to a manyserver queue.The idea of using a continuous distribution for randomlyassigning priority levels was also recently proposed as ascheduling mechanism for the
M/G/ queue [10]. Althoughthe preemptive priority scheduling mechanism is the samefor both our work and the work in [10], a major differenceis that our work (and our previous work [9]) provides a characterization of the distribution of customer priority levelsin the system in equilibrium. In contrast, [10] focuses moreon the effect of the randomized scheduling on the overallpopulation. Another major difference is that we do not assumethat the system is stable. Our current work considers a systemwith many servers and is hence distinct from both [9] and[10].Because of the complexity that arises due to having acontinuum of priority levels, we opt to simplify other aspectsof the model. We note that all customers in our modelexperience the same service rate regardless of their prioritylevel. This differs from other priority queueing models, e.g.[11], and restricts our attention to models in which prioritylevels only impact scheduling and not service rate. We alsofocus on preemptive scheduling as in [12] rather than non-preemtive scheduling as in [13]. By focusing on preemptivescheduling we know that the customer who is being servicedis always the customer with the highest priority. Both of theseassumptions (uniform service rate and preemptive scheduling)were also exploited in [9] and [10].We note that the use of infinite dimensional stochasticprocesses is itself not novel to queueing. Measure-valuedprocesses have been used to study the earliest-deadline-firstdiscipline [14] and the processor-sharing discipline [15], aswell as many server [16] and infinite server models [17]. Inthese contexts, the state of the system varies continuouslyas the dynamic properties of the jobs change. In our model,the priority levels are static and so the state only changesat arrival and departure events. Consequently, our model issubstantially more tractable. Indeed, while these other worksfocus on diffusion approximations, we will only present exactresults.With this motivation and background in mind, the remainderof the paper is organized as follows. In Section II we fullydescribe our model and discuss different choices for the state.In Section III we analyze the steady state behavior of thesystem. We compute the measure that tells us the averagedistribution of customer priority levels in the system. Wederive formulae for the expected sojourn and waiting timesof a customer as a function of his priority level. We note howthese results generalize our previous work [9]. In Section IVwe provide some simulation results that verify our analyticalresults. In Section V we discuss potential future work and weconclude in Section VI. a r X i v : . [ c s . PF ] D ec I. M
ODEL F ORMULATION
In this section we formally describe our model and explainour modeling assumptions. We highlight the fact that certainseemingly limiting assumptions are actually without loss ofgenerality. We present three state representations and explaintheir equivalence. This model is very similar to the model fromour previous work [9]; the key difference is that here we allowfor more than one server.We consider an infinite buffer queue with c servers. Cus-tomers arrive according to a Poisson process with rate α > .Customers have independent and identically distributed (IID)service times that are exponentially distributed. Since timecan be scaled arbitrarily, we assume that the service timeshave unit mean. Therefore, the load is α/c . Customers arealso assigned IID priority levels that are uniformly distributedon the unit interval. The priority levels are independent of allother random quantities in the model. Customers are scheduledpreemptively according to their priorities. When there are atmost c customers in the system, each customer is assigned to aserver; when there are more than c customers in the system, the c with the highest priority levels are assigned to the c serverswhile the rest wait. When a new customer arrives and noservers are available, he may immediately preempt the lowestpriority customer who is in service. The preempted customerwaits in the buffer. In summary, we have an M/M/c queue(not necessarily stable) in which customers are preemptivelyscheduled according to exogenously assigned IID U ([0 , priority levels.Note that customers are scheduled based on their relativeorder rather than their absolute value and consequently, thefact that the priority levels are drawn from U ([0 , (asopposed to some other distribution) is actually without lossof generality. Because the scheduling decisions only dependon the relative order of the priority levels, the dynamics wouldbe unchanged if the priorities were transformed by any order-preserving map. In particular, suppose we want to considerpriority levels that are drawn from some other distributionwith cumulative distribution function (CDF) F ( · ) . Considertwo distinct customers i and j with priority levels p i and p j drawn from U ([0 , . Consider the transformed priority levels ˜ p i = F − ( p i ) and ˜ p j = F − ( p j ) where F − ( · ) is the quantilefunction associated with F ( · ) : F − ( p ) = inf { x ∈ R : p ≤ F ( x ) } (1)If p i > p j then ˜ p i ≥ ˜ p j and we also have that ˜ p i and ˜ p j aredistributed according to the CDF F ( · ) [18, Theorem 2.1]. Soif F ( · ) is strictly increasing then using ˜ p i and ˜ p j yields thesame scheduling dynamics as using p i and p j . If F ( · ) is notstrictly increasing, then with non-zero probability we couldhave ˜ p i = ˜ p j . However, in this situation customers i and j areindistinguishable and these ties can be broken in an arbitraryfashion, e.g. randomly. Consequently, our model encompassesarbitrary distributions of priority levels. For simplicity, we willfocus having priority levels drawn from U ([0 , .We also note that because of the memorylessness propertyof the exponential distribution, if a customer is preempted then his residual service time is still exponentially distributed withunit mean. As a result, any choice for the state does not needto include the residual service time of each customer in thesystem, merely the priority level of each customer. Since thepriority levels are drawn from a continuum, almost surely notwo customers will have the same priority. Consequently, thestate needs to encode the unique priority level of each customerin the system. We find it convenient to encode this list ofpriority levels as a point measure on [0 , . Let B ([0 , bethe σ -algebra of Borel sets on [0 , . Given B ∈ B ([0 , let x t ( B ) be the number of customers in the system at time t with priority levels contained in B . To write this symbolically,let δ z denote a Dirac measure at z ∈ [0 , . If there are N customers in the system at time t and their priority levels are { p , . . . , p N } ⊂ [0 , , then x t can be written as a sum ofDirac measures: x t = N (cid:88) i =1 δ p i (2)Now consider the (non-normalized) CDF or the comple-mentary CDF: X t ( p ) = x t ([0 , p ]) , ¯ X t ( p ) = x t (( p, (3)These two function-valued stochastic processes are actuallyequivalent to the measure-valued process defined above. Theequivalence follows from the fact that { [0 , p ] : p ∈ [0 , } and { ( p,
1] : p ∈ [0 , } each form π -systems that generate B ([0 , . We know that x t ( · ) is finite because it is a countingmeasure. Hence, an elementary application the π - λ Theoremshows that { X t ( p ) : p ∈ [0 , } and (cid:8) ¯ X t ( p ) : p ∈ [0 , (cid:9) eachuniquely define x t ( · ) . The definitions of π -systems and λ -systems along with the method of uniquely extending ameasure from a π -system to a σ -algebra are standard; see [19,Chapter 3] for details.III. S OME T HEORETICAL R ESULTS
We now analyze the equilibrium behavior of the system.First we characterize the steady state distribution of ¯ X t ( p ) for each p ∈ [0 , . We provide a corollary that partiallycharacterizes the steady state distribution of x t ( · ) . We thenprovide formulae for the expected sojourn time and the ex-pected waiting time of a customer as functions of its prioritylevel. Each of these results generalizes our previous resultsregarding the single server case [9]. As in the previous section,we rely on standard results regarding the extension of measuresfrom π -systems to σ -algebras [19, Chapter 3]. Theorem 1.
Fix any p ∈ [0 , , ¯ X t ( p ) converges weakly to arandom variable ¯ X ( p ) . If (1 − p ) α < c , then ¯ X ( p ) has thefollowing probability mass function (PMF) on the non-negativeintegers: P ( ¯ X ( p ) = k ) (4) = (cid:104)(cid:80) c − i =0 ((1 − p ) α ) i i ! + ((1 − p ) α ) c c ! × (1 − (1 − p )( α/c )) (cid:105) − , k = 0 P ( ¯ X ( p ) = 0) × ((1 − p ) α ) k k ! , ≤ k ≤ c P ( ¯ X ( p ) = 0) × ((1 − p ) α ) k c ! × c k − c , k > c f (1 − p ) α ≥ c , then ¯ X ( p ) = ∞ almost surely.Proof. As in [9], the key is to notice that because of the pre-emptive scheduling, the customers with priority levels in ( p, are not affected in any way by customers with priority levelsin [0 , p ] . Moreover, because the priority levels are independentof the inter-arrival times, the customers with priority levels in ( p, arrive according to a Poisson process with rate (1 − p ) α .As a result, ¯ X t ( p ) is stochastically equivalent to the populationin an M/M/c queue with unit rate servers and arrival rate (1 − p ) α . As a result, ¯ X t ( p ) converges weakly to a randomvariable with the given PMF [20, Chapter 3]. Because there isno upper bound on α , it is possible that (1 − p ) α ≥ c . In thiscase, the equivalent M/M/c queue is not stable and hence ¯ X t ( p ) diverges to infinity. Remark 1.
When c = 1 and (1 − p ) α < c we have that P ( ¯ X ( p ) = k ) = (1 − (1 − p ) α )((1 − p ) α ) k (5) for all non-negative integers k . In other words, ¯ X ( p ) is ageometric random variable on the non-negative integers withmean (1 − p ) α/ (1 − (1 − p ) α ) . Hence, this result generalizesour previous work [9, Theorem 1]. Definition 1.
For convenience, we define P ( p ) = P ( ¯ X ( p ) = 0)= (cid:34) c − (cid:88) i =0 ((1 − p ) α ) i i ! + ((1 − p ) α ) c c ! × (1 − (1 − p )( α/c )) (cid:35) − (6) when (1 − p ) α < c . We also define p ( p ) = − ddp P ( p ) (7) = − P ( p ) (cid:34) c − (cid:88) i =1 i (1 − p ) i − α i i ! + c (1 − p ) c − α c c !(1 − (1 − p )( α/c ))+ (1 − p ) c α c +1 c × c !(1 − (1 − p )( α/c )) (cid:35) Corollary 1.
Fix B ∈ B ([0 , . Then x t ( B ) converges weaklyto a random variable x ( B ) with mean µ ( B ) = E [ x ( B )] = (cid:90) B m ( p ) dp (8) where m ( p ) = α + (9) α c +1 c × c ! (cid:34) ( c + 1)(1 − p ) c P ( p ) + (1 − p ) c +1 p ( p )(1 − (1 − p )( α/c )) + 2(1 − p ) c +1 P ( p )( α/c )(1 − (1 − p )( α/c )) (cid:35) when (1 − p ) α < c and m ( p ) = ∞ otherwise. Proof. We can use the PMF from the previous theorem toshow that E [ ¯ X ( p )] = (1 − p ) α + α c +1 (1 − p ) c +1 P ( p ) c × c ! × (1 − (1 − p )( α/c )) . (10)This is the average number of customers in an M/M/c queue with arrival rate (1 − p ) α and unit service rate.Therefore, if B = [ a, b ] for some ≤ a < b ≤ , then x t ( B ) = ¯ X t ( a ) − ¯ X t ( b ) . Since x t ([ a, b ]) converges weaklyto ¯ X ( a ) − ¯ X ( b ) , performing the integration gives us the sameresult subtracting E [ ¯ X ( b )] from E [ ¯ X ( a )] . Indeed, note that for p such that (1 − p ) α < c , m ( p ) = − ddp E [ ¯ X ( p )] . (11)Now note that intervals of this form are a π -system thatgenerates B ([0 , . Consequently, if α < c then µ ([0 , < ∞ and so this defines a unique measure on B ([0 , . On the otherhand, if α ≥ c , we can still extend the measure from the π -system to B ([0 , , but uniqueness is no longer guaranteed.However, we can apply the same reasoning as above to define aunique measure on B ([1 − c/α, where µ ( · ) is finite. The factthat µ ( B ) = ∞ for any B such that B ∩ [0 , − c/α ] has non-zero Lebesgue measure follows from the instability argumentin the previous theorem. Hence, regardless of the value of α ,we can conclude that the expression for the mean equilibriumbehavior of x t ( B ) holds for any B ∈ B ([0 , . Remark 2.
When c = 1 and (1 − p ) α < c we have that m ( p ) = α (1 − (1 − p ) α ) (12) so the corollary generalizes the results in our previous work[9]. Because service can be preempted and hence customers canenter service multiple times, we formally define the terms“sojourn time” and “waiting time”. In particular, we note thatthe amount of time a customer spends in service before beingpreempted is considered waiting. We used the same definitionsin our prior work [9].
Definition 2.
A customer’s sojourn time is the amount oftime from when the customer arrives to when it departs aftercompleting service.
Definition 3.
A customer’s waiting time is the amount of timefrom when the customer arrives to the beginning of the lasttime the customer enters service.
Theorem 2.
Fix any p ∈ [0 , and let s ( p ) be the expectedsojourn time for a customer with priority p in steady state.Then if (1 − p ) α < c then s ( p ) = 1 α m ( p ) (13) and if (1 − p ) α ≥ c then s ( p ) = ∞ .Proof. The case for which s ( p ) = ∞ follows trivially fromthe instability argument in Theorem 1.or the nontrivial case, we first consider ¯ S ( p ) , the averagesojourn time for all customers with priority levels in ( p, .The law of total probability tells us that ¯ S ( p ) = (cid:90) p s ( q ) 11 − p dq. (14)Now we apply Little’s Law [21]. Customers with prioritylevels in ( p, arrive at a rate (1 − p ) α so we have that E [ ¯ X ( p )] = (1 − p ) α ¯ S ( p ) = α (cid:90) p s ( q ) dq. (15)The corollary gives us a formula for E [ ¯ X ( p )] : (cid:90) p s ( q ) dq = 1 α E [ ¯ X ( p )] = 1 α (cid:90) p m ( q ) dq (16)Since this holds for any p , we have that s ( p ) = m ( p ) /α . Corollary 2.
Fix any p ∈ [0 , and let w ( p ) be the expectedwaiting time for a customer with priority p in steady state toreceive service. If (1 − p ) α < c then w ( p ) = s ( p ) − α m ( p ) − (17) and if (1 − p ) α ≥ c then w ( p ) = ∞ .Proof. The sojourn time is the sum of the waiting time andthe service time. Since we have a unit service rate, we merelysubtract 1 from s ( p ) to get w ( p ) . Remark 3.
We note that the relationships between m ( · ) , s ( · ) ,and w ( · ) are the same as they were in the single servercase [9]. Consequently, the previous theorem and corollarygeneralize the results from our previous work. Remark 4. If α ≥ c then m ( · ) (and hence both s ( · ) and w ( · ) )exhibit a bifurcation, i.e. a qualititative change in behavior, at p ∗ = 1 − cα . (18) It is intuitive that when the system is overloaded, lower prioritycustomers will be ignored so that higher priority customerscan be served. The quantity p ∗ makes this intuition precise:when the system is overloaded, customers with priority levelsin [0 , p ∗ ] will have infinite expected waiting times whilecustomers in ( p ∗ , will have finite expected waiting times. Remark 5.
The aforementioned birfurcation makes the caseof α = c particularly interesting. We know that when α = c the M/M/c is unstable. However, in this case p ∗ = 0 so allcustomers with priority levels in (0 , have a finite sojourntime while only customers with priority levels equal to zerohave infinite sojourn times. This seems a bit paradoxical:the queue is unstable but almost every customer has a finitesojourn time. This counterintuitive result arises because α = c is the critical point between a stable M/M/c queue and anunstable
M/M/c queue.
Remark 6.
The previous remarks highlight the fact that thisinfinite dimensional priority scheduling scheme can be used to “partially stabilize” an unstable single class queueing systemin the following sense. If we have a single class
M/M/c system with α ≥ c that is scheduled in either a last-come-first-serve (LCFS) or first-come-first-serve (FCFS) manner, thenwe know that the overall population of the queue will beunstable and we cannot provide any guarantee of reasonableservice to any of the customers. If we instead randomly assignpriority levels to arriving customers and schedule preemptivelyaccording to these priority levels, then we can guarantee that c/α of the customers can expect to have finite waiting times.Moreover, upon arrival we can say with certainty exactlywhich customers will have this guarantee. IV. S
IMULATION V ERIFICATION
In this section, we report the results of two discrete eventsimulations of the system: one with α < c and one with α ≥ c .In both cases, we use the simulated data to estimate m ( · ) , s ( · ) , and w ( · ) . In general, we see that the estimates match ourtheoretical results, thus supporting our analysis. A. Estimation Methods
For each of the functions that we estimate, we first get localestimates and we then linearly interpolate to estimate the entirefunction. The details for each function are outlined below andare the same as in our previous work [9]. For all functions,we assume a discretization of < δ < with an integer N δ = δ − .We compute our estimate of m ( · ) , which we denote ˆ m ( · ) ,as follows:1) Because “Poisson Arrivals See Time Averages” [22], werecord x t ( · ) as observed immediately before each arrival.2) For p i ∈ { δ/ iδ } N δ − i =0 , we average the number ofcustomers with priority levels in the half-open interval [ p i − δ/ , p i + δ/ across our observations. We scalethis average by N δ to get ˆ m ( p i ) .3) We linearly interpolate { ˆ m ( p i ) } N δ − i =0 to get ˆ m ( · ) .We compute our estimate of s ( · ) , which we denote ˆ s ( · ) , ina similar fashion:1) We record the arrival time, the departure time, and thepriority level of each customer. If a customer does notdepart in the time horizon, then his departure time isinfinite.2) For p i ∈ { δ/ iδ } N δ − i =0 , we average the sojourntimes for customers with priority levels in the half-openinterval [ p i − δ/ , p i + δ/ . This gives us ˆ s ( p i ) .3) We linearly interpolate { ˆ s ( p i ) } N δ − i =0 to get ˆ s ( · ) .We compute our estimate of w ( · ) , which we denote ˆ w ( · ) ,in a similar fashion:1) We record the arrival time, the last time that the customerenters service before departing, and the priority level ofeach customer. If the customer never departs then thedeparture time is infinite.2) For p i ∈ { δ/ iδ } N δ − i =0 , we average the waitingtimes for customers with priority levels in the half-openinterval [ p i − δ/ , p i + δ/ . This gives us ˆ w ( p i ) .3) We linearly interpolate { ˆ w ( p i ) } N δ − i =0 to get ˆ w ( · ) . . Estimation Results We use δ = 0 . and a time horizon of T = 2 × . We fix c = 2 servers and consider two values of α . When α = 1 . wehave a stable system and when α = 5 . we have an unstablesystem.First we consider the stable case in which m ( · ) , s ( · ) , and w ( · ) are finite. The results are plotted in Fig. 1. Though abit “noisy”, the estimates generally agree with our theoreticalanalysis. Moreover, we see that the estimates have roughlythe same shape and merely differ by constant factors. Thisconfirms our previous analysis regarding the mean equilibriumbehavior of x t ( · ) , the expected sojourn time, and the expectedwaiting time.Now consider the unstable case for which m ( · ) , s ( · ) , and w ( · ) are finite only for p ∈ ( p ∗ ,
1] = (0 . , . As a result, wedo not plot the functions for p < p ∗ . Because of the verticalasymptote at p ∗ , we use a log-scale for the vertical axis. Theresults are plotted in Fig. 2. In Fig. 2a, we see that ˆ m ( · ) and m ( · ) seem to agree on ( p ∗ , . Fig. 2a also depicts thebifurcation at p ∗ . We see that ˆ m ( p ∗ − δ/ is roughly 10 timesthe value of ˆ m ( p ∗ + δ/ . This reflects the fact that ˆ m ( p ) willdiverge to infinity as T ↑ ∞ for p < p ∗ . We see similar resultsregarding ˆ s ( · ) in Fig. 2b. For p ∈ ( p ∗ , , ˆ s ( p ) and s ( p ) agree.Note that for p < p ∗ , neither ˆ s ( p ) nor s ( p ) appear on the plotbecause both quantities are infinite. Hence, we see that ˆ s ( · ) and s ( · ) agree for all p ∈ [0 , . We see the same results for ˆ w ( · ) : the estimate agrees with the analytic result where bothare finite and also where both are infinite.V. F UTURE W ORK
Our work points to several potential directions of futurework. One is to derive more results about the current model.For example, it would be interesting to know more about thehigher order statistics of x ( · ) . It would also be interestingto extend this model to a network setting. With a singlequeue, x t ( · ) is a point measure on [0 , but for a systemwith n queues we would need to have x t ( · ) be a pointmeasure on [0 , n . It seems reasonable to expect that thesteady state distribution would have a product-form as inJackson’s Theorem [23], but the details of the analysis are notimmediately clear. In particular, although the arriving prioritylevels are IID U ([0 , we need to know how customers’priority levels are correlated after they depart.As noted in our previous work [9], it may also be interestingto consider a heavy traffic analysis. Priority queues are anexample of a system that exhibits “state-space collapse” inheavy traffic [24]. In brief, we would see that upon appropriaterescaling, the diffusion limit associated with X t ( p ) for p < p ∗ would be zero. However, it may be possible to consider adiffusion limit for which p ∗ ↓ so that the diffusion limitdoes not collapse to zero. This idea is not yet well developedbut since our analysis applies to overloaded queues, it mayfruitful to consider. . . . . . . p ˆ m ( p ) m ( p ) (a) ˆ m ( · ) vs. m ( · ) . . . . . . p ˆ s ( p ) s ( p ) (b) ˆ s ( · ) vs. s ( · ) . . . . . . p ˆ w ( p ) w ( p ) (c) ˆ w ( · ) vs. w ( · ) Fig. 1. Estimates of m ( · ) , s ( · ) , and w ( · ) based on the data generated bysimulating the system with c = 2 and α = 1 . . For these values of c and α ,the queue is stable and so we use a linear scale for both axes. . . . . . . p ˆ m ( p ) m ( p ) p ∗ (a) ˆ m ( · ) vs. m ( · ) . . . . . . p ˆ s ( p ) s ( p ) p ∗ (b) ˆ s ( · ) vs. s ( · ) . . . . . . p − − ˆ w ( p ) w ( p ) p ∗ (c) ˆ w ( · ) vs. w ( · ) Fig. 2. Estimates of m ( · ) , s ( · ) , and w ( · ) based on the data generated bysimulating the system with c = 2 and α = 5 . . Because the queue is unstable,the values of the functions become quite large and hence we opt to use alogarithmic vertical axis. VI. C
ONCLUSIONS
We have presented an infinite dimensional model for a manyserver priority queue in which customers are scheduled pre-emptively according to priority levels that are drawn from ancontinuous probability distribution. Our steady state analysischaracterizes the first-order statistics of the measure-valuedprocess that describes the priority levels of the customers in thequeue. We have used derived formulae for the expected sojournand waiting times of a function of customer priority level.These results generalize our previous work [9] and contributeto a broader literature on preemptive scheduling with randompriorities [10]. Discrete event simulations verify our analyticalresults and we have discussed some areas of future work.R
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