Control Policies Approaching HGI Performance in Heavy Traffic for Resource Sharing Networks
aa r X i v : . [ m a t h . P R ] O c t CONTROL POLICIES APPROACHING HGI PERFORMANCE IN HEAVY TRAFFIC FORRESOURCE SHARING NETWORKS
AMARJIT BUDHIRAJA AND DANE JOHNSON A BSTRACT . We consider resource sharing networks of the form introduced in the work of Massouliéand Roberts(2000) as models for Internet flows. The goal is to study the open problem, formulatedin Harrison et al. (2014), of constructing simple form rate allocation policies for broad families ofresource sharing networks with associated costs converging to the Hierarchical Greedy Ideal perfor-mance in the heavy traffic limit. We consider two types of cost criteria, an infinite horizon discountedcost, and a long time average cost per unit time. We introduce a sequence of rate allocation controlpolicies that are determined in terms of certain thresholds for the scaled queue length processesand prove that, under conditions, both type of costs associated with these policies converge in theheavy traffic limit to the corresponding HGI performance. The conditions needed for these resultsare satisfied by all the examples considered in Harrison et al. (2014).
1. I
NTRODUCTION
In [9] the authors have formulated an interesting and challenging open problem for resourcesharing networks that were introduced in the work of Massoulié and Roberts [14] as models forInternet flows. A typical network of interest consists of I resources (labeled 1, . . . , I ) with associatedcapacities C i , i =
1, . . . , I . Jobs of type 1, . . . , J arrive according to independent Poisson processeswith rates depending on the job-type and the job-sizes of different job-type are exponentially dis-tributed with parameters once more depending on the type. Usual assumptions on mutual inde-pendence are made. The processing of a job is accomplished by allocating a flow rate to it over timeand a job departs from the system when the integrated flow rate equals the size of the job. A typicaljob-type requires simultaneous processing by several resources in the network. This relationshipbetween job-types and resources is described through a I × J incidence matrix K for which K i j = j -th job-type requires processing by resource i and K i j = x = ( x , . . . , x J ) ′ the vector of flow rates allocated to various job-types at any given time instant, x must satisfy thecapcity constraint K x ≤ C , where C = ( C , . . . , C I ) ′ .One of the basic problems for such networks is to construct “good” dynamic control policiesthat allocate resource capacities to jobs in the system. A “good” performance is usually quanti-fied in terms of an appropriate cost function. One can formulate an optimal stochastic controlproblem using such a cost function, however in general such control problems are intractable andtherefore one considers an asymptotic formulation under a suitable scaling. The paper [9] formu-lates a Brownian control problem (BCP) that formally approximates the system manager’s controlunder heavy traffic conditions. Since finding optimal solutions of such general Brownian controlproblems and constructing asymptotically optimal control policies for the network based on such D EPARTMENT OF S TATISTICS AND O PERATIONS R ESEARCH , 304 H
ANES H ALL , U
NIVERSITY OF N ORTH C AROLINA ,C HAPEL H ILL , NC 27599
E-mail addresses : [email protected], [email protected] .2010 Mathematics Subject Classification.
Primary: 60K25, 68M20, 90B36. Secondary: 60J70.
Key words and phrases.
Stochastic networks, dynamic control, heavy traffic, diffusion approximations, Browniancontrol problems, reflected Brownian motions, threshold policies, resource sharing networks, Internet flows. BUDHIRAJA AND JOHNSON solutions is a notoriously hard problem, the paper [9] proposes a different approach in which thegoal is not to seek an asymptotically optimal solution for the network but rather control policiesthat achieve the so called
Hierarchical Greedy Ideal (HGI) performance in the heavy traffic limit.Formally speaking, HGI performance is the cost associated with a control in the BCP (which is ingeneral sub-optimal), under which (I) no resource’s capacity is underutilized when there is work forthat resource in the system, and (II) the total number of jobs of each type at any given instant is theminimum consistent with the vector of workloads for the various resources. Desirability of suchcontrol policies has been argued in great detail in [9] through simulation and numerical examplesand will not be revisited here.The main open problem formulated in [9] is to construct simple form rate allocation policies forbroad families of resource sharing networks with associated costs converging to the HGI perfor-mance determined from the corresponding BCP. The goal of this work is to make progress on thisopen problem. We consider two types of cost criteria, the first is an infinite horizon discountedcost (see (2.7)) and the second is a long time average cost per unit time (see (2.8)). In particular thesecond cost criterion is analogous to the cost function considered in [9]. We introduce a sequenceof rate allocation control policies that are determined in terms of certain thresholds for the scaledqueue length processes and prove in Theorems 4.7 and 4.8 that, under conditions, the costs (2.8)and (2.7) associated with these policies converge in the heavy traffic limit to the corresponding HGIperformance.We now comment on the conditions that are used in establishing the above results. The firstmain condition (Condition 2.1) we need is the existence of local traffic on each resource, namelyfor each resource i there is a unique job type that only uses resource i . This basic condition, first in-troduced in [11], is also a key assumption in [9] and is needed in order to ensure that the state spaceof the workload process is all of the positive orthant (see Section 3 for a discussion of this point).Our second condition (Condition 2.2) is a standard heavy traffic condition and a stability conditionfor diffusion scaled workload processes. The stability condition will be key in Section 8.2 when es-tablishing moment bounds that are uniform in time and scaling parameter. We now describe thefinal main condition used in this work. In Section 4 we will see that the collection of all job-typescan be decomposed into the so called primary jobs and secondary jobs. Primary jobs are thosewith ‘high’ holding cost and intuitively are the ones we want to process first. It will also be seen inSection 4 that the collection S of all job-types that only require processing from a single resourceis contained in the collection S s of all secondary jobs. Our third main condition, formulated asCondition 4.4, says that there is a ranking of all job-types in S m . = S s \ S . A precise notion of aranking is given in Definition 4.3, but roughly speaking, the job-types with larger rank value will gethigher ‘attention’ in a certain sense under our proposed policy. We note that the ranking is giventhrough a deterministic map that only depends on system parameters and not on the state of thesystem. The condition is somewhat nontransparent and notationally cumbersome and so we pro-vide two sufficient conditions in Theorems 5.1 and 5.3 for Condition 4.4 to hold. We also discussin Remarks 5.2 and 5.4 some examples where one of these sufficient conditions holds. In partic-ular, all the examples in [9] (2LLN, 3LLN, C3LN, and the negative example of Section 13 therein)satisfy Condition 4.4. Furthermore, there are many other networks not covered by Theorems 5.1and 5.3 where Condition 4.4 is satisfied and in Example 5.5 we provide one such example. Finally,it is not hard to construct examples where Condition 4.4 fails and in Example 5.6 we give such anexample. Construction of simple form rate allocation policies that achieve HGI performance inthe heavy traffic limit for general families of models as in Example 5.6 remains a challenging openproblem. We expect that suitable notions of state dependent ranking maps will be needed in orderto use the ideas developed in the current work for treating such models, however the proofs andconstructions are expected to be substantially more involved. ONTROL POLICIES FOR HGI PERFORMANCE 3
Our rate allocation policy is introduced in Definition 4.5. Implementation of this policy requiresfirst determining the collection of secondary jobs. This step, using the definition in (4.1), can becompleted easily by solving a finite collection of linear programming problems. The next step isto determine a viable ranking (if it exists) of all jobs in S m . In general when S m is very large, de-termining this ranking may be a numerically hard problem, however as discussed in Section 5.1,for many examples this ranking can be given explicitly in a simple manner. Once a ranking is de-termined, the policy in Definition 4.5 is explicit given in terms of arbitrary positive constants c , c with c < c and α ∈ (0, 1/2). Roughly speaking, our approach is applicable to systems where job-types have a certain ordering of “urgency” in the sense that, regardless of the particular workload,we want as much of it as possible to come from the least urgent job types. A second concern thatneeds to be addressed is that a resource should work at ‘near’ full capacity when there is ‘non negli-gible’ amount of work for it. A detailed discussion of how the proposed policy achieves these goalsis given in Remark 4.6 where we also comment on connection between this policy and the UFOpolicies proposed in [9].We now comment on the proofs of our main results, Theorems 4.7 and 4.8. Both results rely onlarge deviation probability estimates and stopping time constructions of the form introduced firstin the works of Bell and Williams [2, 3] (see also [5] and [1]). A key result is Theorem 8.2 whichrelates the cost under our policy with the workload cost function C in (3.1). This estimate is crucialin achieving property (II) of the HGI asymptotically. Asymptotic achievement of property (I) of HGIis a consequence of Theorem 9.2, the estimate in (10.1) and continuity properties of the Skorohodmap. Proof of Theorem 4.7 requires additional moment estimates that are uniform in time and thescaling parameter (see Section 8.2). A key such estimate is given in Theorem 8.5, the proof of whichrelies on the construction of a suitable Lyapunov function (see Proposition 8.8). Once uniformmoment bounds are available, one can argue tightness of certain path occupation measures (seeTheorem 9.3) and characterize their limit points in a suitable manner (see Theorem 9.4). Desiredcost convergence then follows readily by appealing to continuous mapping theorem and uniformintegrability estimates.The paper is organized as follows. In Section 2 we introduce the state dynamics, cost functionsof interest, and two of our main conditions. Section 3 gives the precise definition of HierarchicalGreedy Ideal Performance in terms of certain costs associated with I dimensional reflected Brown-ian motions. In Section 4 we introduce our final key condition (Condition 4.4), present our dynamicrate allocation policy, and give our two main convergence results: Theorems 4.7 and 4.8. Section 5discusses Condition 4.4 and presents some sufficient conditions for it to be satisfied. This sectionalso gives an example where the condition fails to hold. Sections 6 - 9 form the technical heart ofthis work. Section 6 proves some useful properties of the workload cost function C ( · ) introducedin (3.1) and Section 7 studies some important structural properties of our proposed rate allocationpolicy. Section 8 is technically the most demanding part of this work. It provides some key esti-mates on costs under our scheme in terms of the workload cost function and establishes certainmoment estimates that are uniform in time and the scaling parameter. In Section 9 we introducecertain path occupation measures, prove their tightness, and characterize the limit points. FinallySection 10 completes the proof our two main results. An appendix contains some standard largedeviation estimates for Poisson processes.The following notation will be used. For a Polish space S , denote the corresponding Borel σ -fieldby B ( S ). Denote by P ( S ) (resp. M ( S )) the space of probability measures (resp. finite measures)on S , equipped with the topology of weak convergence. For f : S → R , let k f k ∞ . = sup x ∈ S | f ( x ) | .For a Polish space S and T >
0, denote by C ([0, T ] : S ) (resp. D ([0, T ] : S )) the space of continuousfunctions (resp. right continuous functions with left limits) from [0, T ] to S , endowed with the uni-form topology (resp. Skorokhod topology). We say a collection { X n } of S -valued random variables BUDHIRAJA AND JOHNSON is tight if the distributions of X n are tight in P ( S ). Equalities and inequalities involving vectors areinterpreted component-wise. 2. G ENERAL B ACKGROUND
Assume there are J types of jobs and I resources for processing them. The network is describedthrough the I × J matrix K that has entries K i j = i works on job type j , and K i , j = K are identical, namely, givena subset of resources, there is at most one job-type that has this subset as the associated set ofresources. Given m ∈ N , we let N m . = {1, 2, . . . m }. In particular, N I = {1, . . . I } and N J = {1, . . . J }.Denote by N j the set of resources that work on type j jobs, i.e. N j . = { i ∈ N I : K i , j = S be the collection of all job types that use only one resource. I.e. S . = { j ∈ N J : T K e j = I X i = K i j = e j is the unit vector in R J with 1 in the j -th coordinate and is the I -dimensional vector ofones. Throughout we assume that for every resource there is a unique job type that only uses thatresource, namely the following condition is satisfied. Condition 2.1. S j ∈ S N j = N I We denote the unique job-type that uses only resource i as ˇ j ( i ). Similarly for j ∈ S , we denoteby ˆ i ( j ) the unique resource that processes this job-type.The capacity for resource i is given by C i . Let { η rj ( k )} ∞ k = be the i.i.d. inter-arrival times for the j -th job type and let { ∆ rj ( k )} ∞ k = be the associated i.i.d. amounts of work for the j -th job type. Ifat a given instant work of type j is processed at rate x j then the capacity constraint requires that C ≥ K x . We assume the { η rj ( k )} ∞ k = are exponentially distributed with rates λ rj and the { ∆ rj ( k )} ∞ k = are exponentially distributed with rates µ rj . Define Poisson processes A rj ( t ) = max ( k : k X i = η rj ( i ) ≤ t ) , S rj ( t ) = max ( k : k X i = ∆ rj ( i ) ≤ t ) .Let ̺ rj = λ rj µ rj and ̺ r . = ( ̺ rj ) Jj = . The following will be our main heavy traffic condition. The require-ment v ∗ > Condition 2.2. C > K ̺ r for all r . For some λ j , µ j ∈ (0, ∞ ) , lim r →∞ λ rj = λ j , lim r →∞ µ rj = µ j , for allj ∈ N J . With ̺ j = λ j µ j and ̺ = ( ̺ j ) j ∈ N J , C = K ̺ , lim r →∞ r ( ̺ − ̺ r ) = β ∗ , v ∗ . = K β ∗ > . Consider a J -dimensional absolutely continuous, nonnegative, non-decreasing stochastic pro-cess { B r ( t )} where B rj ( t ) represents the amount of type j work processed by time t under a givenpolicy. Note that such a process must satisfy the resource constraint: K ˙ B rj ( t ) ≤ C , for all t ≥
0. (2.1)Define the I dimensional capacity-utilization process T r = K B r . Then T ri ( t ) represents theamount of work processed by the i -th resource by time t . Letting I r ( t ) = tC − T r ( t ), I ri ( t ) represents ONTROL POLICIES FOR HGI PERFORMANCE 5 the unused capacity of resource i by time t . Let { Q r ( t )} be the J -dimensional process, where Q rj ( t )represents the number of jobs in the queue for type j jobs. Then Q r ( t ) = q r + A r ( t ) − S r ¡ B r ( t ) ¢ , (2.2)where q r denotes the initial queue-length vector. For B r to be a valid rate allocation policy, Q r defined by (2.2) must satisfy Q r ( t ) ≥ t ≥
0. (2.3)Any absolutely continuous, nonnegative, non-decreasing stochastic process { B r ( t )} satisfying (2.1),(2.3) and appropriate non-anticipativity conditions will be referred to as a resource allocation pol-icy or simply a control policy . Non-anticipativity conditions on { B r } are formulated using multi-parameter filtrations as in [6] (see Definition 2.6 (iv) therein). We omit details here, however wewill note that from Theorem 5.4 of [6] it follows that the control policy constructed in Section 4.1 isnon-anticipative in the sense of [6].Let W r ( t ) be the I -dimensional workload process given by W r ( t ) = K M r Q r ( t ) where M r is thediagonal matrix with entries 1/ µ rj .Define the fluid-scaled quantities by¯ T r ( t ) = T r ( r t )/ r , ¯ B r ( t ) = B r ( r t )/ r , ¯ I r ( t ) = I r ( r t )/ r ¯ A r ( t ) = A ( r t )/ r , ¯ S r ( t ) = S r ( r t )/ r ,¯ Q r ( t ) = Q r ( r t )/ r , ¯ W r ( t ) = W ( r t )/ r (2.4)and the diffusion scaled quantitiesˆ T r ( t ) = T ( r t )/ r , ˆ B r ( t ) = B r ( r t )/ r , ˆ I r ( t ) = I r ( r t )/ r ,ˆ A r ( t ) = ( A ( r t ) − λ r r t )/ r , ˆ S r ( t ) = ( S r ( r t ) − µ r r t )/ r ,ˆ Q r ( t ) = Q ( r t )/ r , ˆ W r ( t ) = W r ( r t )/ r . (2.5)Note that, with G r . = K M r and ˆ w r . = G r ˆ q r ,ˆ W r ( t ) = G r ˆ Q r ( t ) = ˆ w r + G r ( ˆ A r ( t ) − ˆ S r ( ¯ B r ( t ))) + t r ( K ̺ r − C ) + r ¯ I r ( t ). (2.6)Let h be a given I -dimensional strictly positive vector. Associated with a control policy B r , We willbe interested in two types of cost structures: • Infinite horizon discounted cost:
Fix θ ∈ (0, ∞ ). J rD ( B r , q r ) . = Z ∞ e − θ t E ¡ h · ˆ Q r ( t ) ¢ d t . (2.7) • Long-term cost per unit time: J rE ( B r , q r ) . = lim sup T →∞ T Z T E ¡ h · ˆ Q r ( t ) ¢ d t . (2.8)The goal of this work is to construct dynamic rate allocation policies that asymptotically achievethe Hierarchical Greedy Ideal(HGI) performance as r → ∞ . The next section gives the precise defi-nition of HGI performance. BUDHIRAJA AND JOHNSON
3. H
IERARCHICAL G REEDY I DEAL
Similar to M r and G r in Section 2, let M be the J × J diagonal matrix with entries {1/ µ j } Jj = andlet G . = K M . Define for w ∈ R I + (regarded as a workload vector), the set of possible associated queuelengths Q ( w ) by the relation Q ( w ) . = { q ∈ R J + : G q = w }.Note that by our assumption on K , Q ( w ) is compact for every w ∈ R I + . Also the local traffic con-dition (Condition 2.1) ensures that Q ( w ) is nonempty for every w ∈ R I + . HGI performance intro-duced in [9] is motivated by the Brownian control problem (BCP), as introduced in [8], associatedwith the network in Section 2 and the holding cost vector h . This BCP has an equivalent workloadformulation (EWF) from the results of [10] (see Section 10 of [9]). The EWF in the current settingis a singular control problem with state space that is all of the positive orthant R I + (due to the localtraffic condition). In the EWF the cost is given by a nonlinear function C defined as C ( w ) . = inf q ∈ Q ( w ) { h · q }, w ∈ R I + . (3.1)One particular control in the EWF is the one corresponding to no-action in the interior and normalreflection on the boundary of the orthant. This control yields the (coordinate-wise) minimal con-trolled state process in the EWF given as the I -dimensional reflected Brownian motion in R I + withnormal reflection. The HGI performance is the cost, in terms of the the workload cost function C ,associated with this minimal state process. We now give precise definitions.We first recall the definition of the Skorohod problem and Skorohod map with normal reflectionson the d -dimensional positive orthant. Definition 3.1.
Let ψ ∈ D ([0, T ] : R d ) such that ψ (0) ∈ R d + . The pair ( ϕ , η ) ∈ D ([0, T ] : R d × R d ) is saidto solve the Skorohod problem for ψ (in R d + , with normal reflection) if ϕ = ψ + η ; ϕ ( t ) ∈ R d + for allt ≥ ; η (0) = ; η is nondecreasing and R [0, T ] { ϕ i ( t ) > d η i ( t ) = . We write ϕ = Γ d ( ψ ) and refer to Γ d as the d -dimensional Skorohod map . It is known that there is a unique solution to the above Skorohod problem for every ψ ∈ D ([0, T ] : R d ) and that the Skorohod map has the following Lipschitz property: There exists K Γ d ∈ (0, ∞ ) suchthat for all T > ψ i ∈ D ([0, T ] : R d ) such that ψ i (0) ∈ R d + , i =
1, 2sup ≤ t ≤ T | Γ d ( ψ )( t ) − Γ d ( ψ )( t ) | ≤ K Γ d sup ≤ t ≤ T | ψ ( t ) − ψ ( t ) | .Also note that for ψ ∈ D ([0, T ] : R d ), Γ d ( ψ ) i = Γ ( ψ i ) for all i =
1, . . . d . When d = I we will write Γ d = Γ I as simply Γ .Let ( ˇ Ω , ˇ F , { ˇ F t }, ˇ P ) be a filtered probability space on which is given a J -dimensional standard{ ˇ F t }- Brownian motion { ˇ B ( t )}. Let ζ j . = ̺ j / µ j for j ∈ J and let Diag( ζ ) be the J × J diagonal matrixwith j -th diagonal entry ζ j . Let Λ . = K ( Diag( ζ )) . For w ∈ R I + , let ˇ W w be a R I + valued continuousstochastic process defined as ˇ W w ( t ) = Γ ( w − v ∗ ι + Λ ˇ B ( · ))( t ), t ≥ ι : [0, ∞ ) → [0, ∞ ) is the identity map. Then ˇ W w is a I -dimensional reflected Brownian mo-tion with initial value w , drift − v ∗ and covariance matrix ΛΛ ′ . It is well known[7] that { ˇ W w } w ∈ R I + defines a Markov process that has a unique invariant probability distribution which we denote as π . ONTROL POLICIES FOR HGI PERFORMANCE 7
Suppose ˆ q r . = q r / r → q as r → ∞ and let w . = G q . Then the HGI cost associated with the costs J rD ( B r , q r ) and J rE ( B r , q r ) are given respectively asHGI D ( w ) . = Z ∞ e − θ t E ¡ C ( ˇ W w ( t )) ¢ d t HGI E . = Z R + I C ( w ) π ( d w ). (3.3)4. C ONTROL P OLICY AND C ONVERGENCE TO
HGIThis section will introduce our final key condition on the model and present our main results.Denote by g , . . . g J the columns of the matrix G , i.e. G = [ g , . . . , g J ]. We will partition the set N J intosets S p and S s corresponding to the set of primary jobs and the set of secondary jobs respectively,defined as follows S p . = { j ∈ N J : C ( g j ) < h j }, S s . = N J \ S p . (4.1)Intuitively, S p corresponds to the set of jobs that we want to process first.Within the set of secondary jobs we will distinguish the set S , introduced earlier, of all job typesthat use only one resource. Note that S is indeed a subset of S s since for j ∈ S , Q ( g j ) = { e j }and so C ( g j ) = inf q ∈ Q ( g j ) { h · q } = h j .We now introduce the notion of minimal covering sets associated with any j ∈ N J and also de-fine, for given F ⊂ N J \ { j }, minimal covering sets of j that are not covering sets for any j ′ ∈ F . Definition 4.1.
Given E ⊂ N J and k ∈ N J we define M E , k to be the collection of all minimal sets ofjobs in E other than k such that N k is contained in the set of all resources associated with the jobs inthe set, namely, M E , k . = ( M ⊂ E \ { k } : N k ⊆ [ j ∈ M N j and N k * [ j ∈ M \{ l } N j for all l ∈ M ) . In addition, given F ⊂ N J define M E , kF to be the collection of all M ∈ M E , k such that the set of re-sources associated with any job in F is not contained in the set of resources associated with jobs inM , namely, M E , kF . = ( M ∈ M E , k : N l * [ j ∈ M N j for any l ∈ F ) .Minimal covering sets will be used to determine the collection of jobs which do not have lowerpriority than any other job in a given subset of N J . For that we introduce the following definition.Let S m . = S s \ S be the collection of secondary jobs that use multiple resources and let m . = | S m | .Denote the j -th column of K by K j , i.e. K = [ K , . . . , K J ]. Definition 4.2.
Given sets E , F ⊂ S m define the set O EF ⊂ E by j ′ ∈ O EF if and only if for all M ∈ M E S S , j ′ F µ j ′ h j ′ + C Ã X j ∈ M K j − K j ′ ! ≤ C Ã X j ∈ M K j ! . (4.2) and the set O E ⊂ E by j ′ ∈ O E if and only if (4.2) holds for all M ∈ M E S S , j ′ . Note that since a M ∈ M E S S , j ′ F covers j ′ , P j ∈ M K j − K j ′ is a nonnegative vector. We now intro-duce the notion of a viable ranking of jobs in S m . BUDHIRAJA AND JOHNSON
Definition 4.3.
A viable ranking of jobs in S m is a bijection ρ : N m → S m , such that for all k ∈ N m , ρ ( k ) ∈ O E k F k , where for k ∈ N m , F k . = { ρ (1), ..., ρ ( k − and E k . = S m \ F k , with the convention that O E k F k = O S m for k = . For an interpretation of a viable ranking, see Remark 4.6. The following will be one of mainassumptions that will be taken to hold throughout this work. This assumption (and Conditions 2.1and 2.2) will not be noted explicitly in the statements of the results.
Condition 4.4.
There exists a viable ranking of jobs in S m . In Section 5 we illustrate through examples that this condition holds for a broad family of mod-els.We can now present our dynamic rate allocation policy.4.1.
Resource Allocation Policy.
For k ∈ N m let ζ ki = { j ∈ N J \ F k + : K i , j = i and have a higher pro-cessing priority than job ρ ( k ) (see Remark 4.7).Let 0 < α < < c < c . Define σ r ( t ) . = n j ∈ N J : Q rj ( t ) ≥ c r α o to be the set of job-types whose queue length is at least c r α at time t . Define ̟ r ( t ) . = [ j ∈ σ r ( t ) N j to be the subset of N I consisting of resources associated with job-types in σ r ( t ), namely with queuelengths at least c r α . We will use the following work allocation scheme. Definition 4.5.
Let δ = min j ̺ j J . For t ≥ , define the vector y ( t ) = ( y j ( t )) j ∈ N J as follows. Primary jobs.
For j ∈ S p y j ( t ) . = ̺ j + δ , if j ∈ σ r ( t ) ̺ j − δ J m + , if j ∉ σ r ( t ). (4.4) Jobs in S m . For k ∈ N m y ρ ( k ) ( t ) . = ̺ ρ ( k ) − k − m − δ , if ζ ki ∩ σ r ( t )
6= ; for all i ∈ N ρ ( k ) ̺ ρ ( k ) + k − m − δ , if ζ ki ∩ σ r ( t ) = ; for some i ∈ N ρ ( k ) and ρ ( k ) ∈ σ r ( t ) ̺ ρ ( k ) − − k − m − δ , if ζ ki ∩ σ r ( t ) = ; for some i ∈ N ρ ( k ) and ρ ( k ) ∉ σ r ( t ). (4.5) Jobs in S . For j ∈ S y j ( t ) . = C ˆ i ( j ) − P l j : K ˆ i ( j ), l = y l ( t ), if ˆ i ( j ) ∈ ̟ r ( t ) ̺ j − δ , if ˆ i ( j ) ∉ ̟ r ( t ). (4.6) For all j , define stopping times τ j = inf{ t ≥ Q rj ( t ) < c r α } , ONTROL POLICIES FOR HGI PERFORMANCE 9 τ j l = inf{ t ≥ τ j l − : Q rj ( t ) ≥ c r α } ,and τ j l + = inf{ t ≥ τ j l : Q rj ( t ) < c r α } ,for all l > . Define E r ( t ) ∈ {0, 1} J by E rj ( t ) . = if t ∈ h τ j l − , τ j l ´ for some l > otherwise.Finally, define x ( t ) ∈ R J as x j ( t ) . = y j ( t )1 { E rj ( t ) = for j ∈ N J . We note that y j ( t ) and x j ( t ) depend on r but this dependence is suppressed in the notation. Remark 4.6.
Roughly speaking, under the allocation policy in Definition 4.5, jobs are prioritizedas follows: S p ≻ S ≻ ρ ( m ) ≻ ρ ( m − · · · ≻ ρ (1). (4.7)However the above priority order needs to be interpreted with some care. We will call the j -thqueue stocked at time instant t if Q rj ( t ) ≥ c r α and we will call it depleted at time instant t if Q rj ( t ) < c r α . The last line of Definition 4.5 says that any queue once depleted does not get anyrate allocation until it gets stocked again. Beyond that, rate allocation by a typical resource i isdecided as follows.First we consider all the primary job-types associated with resource i , i.e. j ∈ S p such that K i j =
1. If the associated queue is stocked then it gets higher than nominal rate allocation according tothe first line in (4.4) and otherwise a lower than nominal allocation as in the second line of (4.4).Next we look at all the job-types in S m associated with resource i . Denote these as j , j , . . . j k and assume without loss of generality that ρ ( j ) < ρ ( j ) · · · < ρ ( j k ). We consider the top ranked job ρ ( j k ) first and look at all the resources (including resource i ) that process this job-type. If everyassociated resource has at least one job-type rated higher according to (4.7) with a stocked queuethen rate allocated to job-type ρ ( j k ) is lower than nominal as given in the first line of (4.5). On theother hand, if there is at least one associated resource such that none of its job-types that are ratedhigher that ρ ( j k ) (according to (4.7)) has a stocked queue , we assign ρ ( j k ) a flow rate higher thannominal, according to the second line in (4.5) if the queue for job-type ρ ( j k ) is stocked and a lowerthan nominal flow rate according to the third line in (4.5) if the queue is not stocked. Note that allresources processing job-type ρ ( j k ) allocate the same flow rate to it. We then successively consider ρ ( j k − ), ρ ( j k − ), . . . ρ ( j ) and allocate rate flows to it in a similar fashion as above.Finally, if the unique job-type ˇ j ( i ) queue associated with resource i is stocked, we allocate it allremaining capacity of resource i (this may be larger or small than nominal allocation) and if thisqueue is not stocked we assign it less than nominal allocation given by the second line in (4.6).Lemma 7.1 will show that B r ( t ) . = R t x ( s ) d s is nonnegative, nondecreasing and satisfies the re-source constraint (2.1). Also, clearly the associated Q r defined by (2.2) satisfies (2.3). Finally, it canbe checked that the process B r ( t ) is non-anticipative in the sense of Definition 2.6 (iv) of [6]. Thus B r is a resource allocation policy as defined in Section 2.We remark that the formal priority ordering given in (4.7) is consistent with the UFO priorityscheme proposed in Section 12 of [9] for 2LLN and 3LLN networks. However, the UFO scheme forC3LN network in [9] appears to be of a different form. Main Results.
Recall that we assume throughout that Conditions 2.1, 2.2 and 4.4 are satisfied.We now present the main results of this work. The first result considers the ergodic cost whereasthe second the discounted cost. Recall q r introduced in (2.2). Theorem 4.7.
Suppose ˆ q r . = q r / r satisfies sup r > ˆ q r < ∞ . Let t r ↑ ∞ as r → ∞ . Then as r → ∞ t r R t r h · ˆ Q r ( t ) d t converges in L to R C ( y ) π ( d y ) . In particular, as r → ∞ ,J rE ( B r , q r ) → HGI E . Theorem 4.8.
Suppose that ˆ q r → q as r → ∞ . Let w = G q . Then lim r →∞ J rD ( B r , q r ) = HGI D ( w ).Proofs of the above theorems are given in Section 10.5. V ERIFICATION OF C ONDITION S s (see Theorem 6.2). For this reason, sufficient conditions below impose condi-tions only on jobs in S s . Finally, for notational convenience, in this section we will denote the jobtype j that requires service from nodes i , ..., i n by χ i ,..., i n . Similarly, we will use notation h χ i in , µ χ i in , and N χ i in for the corresponding h j , µ j , N j .5.1. Some Simple Sufficient Conditions for Condition 4.4.
We present below two basic sufficient(but not necessary) conditions for Condition 4.4 to be satisfied in order to illustrate networks thatare covered by our approach.
Theorem 5.1.
If for all j , k ∈ S m either N j ⊂ N k , N k ⊂ N j , or N j ∩ N k = ; then Condition issatisfied.Proof. We will use the notation from Definition 4.3, namely F k . = { ρ (1), ..., ρ ( k − E k . = S m \ F k .Take ρ to be an arbitrary map from N m to S m with the property that for all j , k ∈ N m with j < k ,either N ρ ( k ) ⊂ N ρ ( j ) or N ρ ( j ) ∩ N ρ ( k ) = ; . Note that our assumption in the statement of the theoremensures that such a map always exists. We now argue that this ρ defines a viable ranking, namelyCondition 4.4 is satisfied. For this we need to show that for every k ∈ N m , ρ ( k ) ∈ O E k F k , namely for all M ∈ M E k S S , ρ ( k ) F k µ ρ ( k ) h ρ ( k ) + C Ã X j ∈ M K j − K ρ ( k ) ! ≤ C Ã X j ∈ M K j ! . (5.1)Now consider such a k and M . Note that M ⊂ { ρ ( k + ρ ( m )} S S . Since M defines a minimalcovering, if for l l ′ , ρ ( l ), ρ ( l ′ ) ∈ M , we must have that N ρ ( l ) ∩ N ρ ( l ′ ) = ; . From minimality of M we also have that, ( S j ∈ S ∩ M N j ) ∩ N ρ ( l ) = ; for every l ≥ k + ρ ( l ) ∈ M . We thus have P j ∈ M | N j | = | N ρ ( k ) | which implies that X j ∈ M K j = K ρ ( k ) . (5.2) ONTROL POLICIES FOR HGI PERFORMANCE 11
Therefore, µ ρ ( k ) h ρ ( k ) + C Ã X j ∈ M K j − K ρ ( k ) ! = µ ρ ( k ) h ρ ( k ) = µ ρ ( k ) C ( g ρ ( k ) ) = C ( µ ρ ( k ) g ρ ( k ) ) = C ( K ρ ( k ) ) = C ( X j ∈ M K j ),where the first and last equality use (5.2) and the second equality uses the fact that ρ ( k ) is a sec-ondary job. This proves (5.1) (in fact with equality) and completes the proof of the theorem. (cid:3) Remark 5.2.
One simple consequence of Theorem 5.1 is that any network where S m = ; (meaning S s = S ) satisfies Condition 4.4. We note that condition S m = ; does not rule out existence ofjobs that require service from multiple nodes. Here is one elementary example to illustrate thispoint. Suppose I = J = µ j = j . Also let h χ = h χ = h χ = h χ = h χ = h χ =
4. It is easy to check that for this example S m = ; .Another consequence of Theorem 5.1 is that any network where S m only contains one job (forinstance a job which impacts all nodes) satisfies Condition 4.4. In particular any 2 node networksatisfies Condition 4.4. Another basic network covered by Theorem 5.1 is one with 2 n jobs where S m = { χ n , χ , χ , χ n − n }. Many other examples can be given. In particular 2LLN and3LLN networks of [9] satisfy the sufficient condition in Theorem 5.1 .The following theorem provides another sufficient condition for a network to satisfy Condition4.4. Recall that O S m is the collection of all j ′ ∈ S m that satisfy (4.2) for all M ⊂ S s \ { j ′ } that areminimal covering sets for j ′ . Theorem 5.3.
If for all j ∈ S m ² O S m and M ∈ M { S m \ O S m } S S , j O S m we have P l ∈ M | N l | = | N j | thenCondition is satisfied.Proof. Consider the following ranking of jobs in S m . Assign the first ¯ m . = ¯¯ O S m ¯¯ ranks arbitrarilyto jobs in O S m and the remaining m − ¯ m ranks arbitrarily to jobs in S m ² O S m . In particular ρ ( k ) ∈ O S m for all k ∈ {1, ..., ¯ m } and ρ ( k ) ∈ S m ² O S m for all k ∈ { ¯ m +
1, ..., m }. Note that, for k ∈ {1, ..., ¯ m }we have O S m ⊂ O E k F k which says that ρ ( k ) ∈ O E k F k for all k ∈ {1, ..., ¯ m }. Let now k ∈ { ¯ m +
1, ..., m }be arbitrary and note that M E k S S , ρ ( k ) F k ⊂ M { S m \ O S m } S S , ρ ( k ) O S m so for all M ∈ M E k S S , ρ ( k ) F k we have P l ∈ M | N l | = | N ρ ( k ) | . This implies that (5.2) is satisfied which as in the proof of Theorem 5.1 showsthat (4.2) is satisfied for all M ∈ M E k S S , ρ ( k ) F k and therefore ρ ( k ) ∈ O E k F k for all k ∈ { ¯ m +
1, ..., m }. Thus ρ defines a viable ranking and so Condition 4.4 is satisfied. (cid:3) Remark 5.4.
The above theorem provides an easy way to check that Condition 4.4 is satisfied.For instance, for 3 node networks if χ ∈ S m , from Theorem 5.3, verification of Condition 4.4reduces to proving that χ ∈ O S m . This is due to the fact that for a 3 node network S m ⊂ { χ , χ , χ , χ }, and consequently for any job j ∈ S m ² { χ } and M ∈ M S m \ { χ } S S , j { χ } wemust have M ∩ S m = ; which says that P l ∈ M | N l | = | N j | . In particular the C3LN in [9] satisfiesthe sufficient condition in Theorem 5.3 with one viable ranking given as ρ (1) = χ , ρ (2) = χ , ρ (3) = χ .Similarly, for a 4 node network with S m ⊂ { χ , χ , χ , χ , χ }, from Theorem 5.3,verification of Condition 4.4 reduces to proving that χ ∈ O S m . Many other examples can begiven. In general Theorem 5.3 can be useful for verifying Condition 4.4 for networks with highnumber of nodes when S m has few elements. In particular the negative example in Section 13of [9] satisfies the sufficient condition in the above theorem. In that example J = I = S m = { χ , χ , χ }. It is easy to see that with the values of holding costs and job sizes in the above paper O S m = { χ , χ } and S m ² O S m = { χ } and so the only M ∈ M { S m \ O S m } S S , j O S m for j = χ is the set { χ , χ } which clearly satisfies the property P l ∈ M | N l | = | N j | .It should be noted that Theorems 5.1 and 5.3 are much more restrictive than necessary, meaningthat the class of networks which satisfy Condition 4.4 is much wider than those covered by Theorem5.1 or Theorem 5.3. To illustrate this we provide a simple example of one such network. Example 5.5.
Let I = J =
7, and µ χ = µ χ = µ χ = µ χ = µ χ = µ χ = µ χ = h χ = h χ = h χ = h χ = h χ = h χ = h χ = S m = { χ , χ , χ } and there is exactly one viable ranking as in Defini-tion 4.3 which is ρ (1) = χ , ρ (2) = χ , ρ (3) = χ (so Condition 4.4 is satisfied). In particularthis implies O S m = { χ }. However, note that N χ N χ , N χ N χ , and N χ ∩ N χ
6= ; so this network does not satisfy the conditions of Theorem 5.1. In addition, χ ∈ S m ² O S m and{ χ , χ } ∈ M { S m \ O S m } S S , χ O S m but | N χ | + | N χ | > | N χ | so the conditions of Theorem 5.3 are notsatisfied either. Consequently this simple network satisfies Condition 4.4 although it is outside thescope of Theorems 5.1 and 5.3.As seen in the last two theorems, Condition 4.4 holds for a broad range of networks. Howeverthere are many interesting cases that are not covered by this condition. We now illustrate this pointthrough an example. In this example I = J = C is a non decreasing function, howevera viable ranking does not exist and therefore techniques of this paper do not apply. Example 5.6. (Example That Doesn’t Satisfy Condition 4.4) Suppose that µ χ = µ χ = µ χ = µ χ = µ χ = µ χ = h χ = h χ = h χ = h χ = h χ = h χ = S m = { χ , χ , χ }. This network does not satisfy Condi-tion 4.4 because O { χ , χ , χ } = ∅ , since (4.2) does not hold for χ , χ , or χ . We leave theverification of this fact to the reader. Consequently a viable ranking cannot exist. Workload cost and its minimizer.
The workload C for this example can be given explicitly asfollows. Let for w ∈ R + , w . = w ∧ w , w . = w ∧ w , w . = w ∧ w ∧ w .For w ∈ R + C ( w ) . = w + w + w , if w ≥ w + w w + w + w , if w + w > w ≥ w ∨ w w + w + w ) + w − w − w , if w ∨ w > w .The optimal q ∗ ( w ) in Q ( w ) is given as follows. Let q ∗ = ( q ∗ χ , q ∗ χ , q ∗ χ , q ∗ χ , q ∗ χ , q ∗ χ ). Then q ∗ ( w ) = (0, w − w − w , 0, w , w , 0), if w ≥ w + w (0, 0, 0, w − w , w − w , w + w − w ), if w + w > w ≥ w ∨ w ( w − w , w + w − w − w , w − w , w − w , w − w , w ), if w ∨ w > w . Note that C and q ∗ are continuous functions and C is nondecreasing. In particular the HGI per-formance in this case is also the optimal cost in the associated BCP. However, as noted above, theredoes not exist a viable ranking for this example. Thus the techniques developed in the currentpaper do not apply to this example. ONTROL POLICIES FOR HGI PERFORMANCE 13
6. S
OME P ROPERTIES OF THE W ORKLOAD C OST F UNCTION
The following result on a continuous selection of a minimizer is well known (cf. Theorem 2 in[4] or Proposition 8.1 in [9]).
Theorem 6.1.
There is a continuous map ¯ q : R I + → R J + such that for every w ∈ R I + , ¯ q ( w ) ∈ Q ( w ) andh · ¯ q ( w ) = C ( w ).Define for a given workload vector w ∈ R I + the set Q s ( w ) consisting of all queue-length vectorsthat produce the workload w and have zero coordinates for queue-lengths corresponding to pri-mary jobs, namely, Q s ( w ) = © q ∈ Q ( w ) : q j = j ∈ S p ª .The following theorem shows that in computing the infimum in (3.1) we can replace Q ( w ) with Q s ( w ). Theorem 6.2.
For all w ∈ R I + , ¯ q ( w ) ∈ Q s ( w ) . In particular, C ( w ) = inf q ∈ Q s ( w ) © h · q ª .Proof. Fix w ∈ R I + . With ¯ q as in Theorem 6.1, we have C ( w ) = h · ¯ q ( w ). Assume ¯ q k ( w ) > k ∈ S p . Then with q ∗ . = ¯ q ( g k ), we have from the definition of S p that h · q ∗ = C ( g k ) < h k . (6.1)Define ˜ q ∈ R J + by ˜ q k = ¯ q k ( w ) q ∗ k and ˜ q j = ¯ q j ( w ) + ¯ q k ( w ) q ∗ j for j k . Then for i ∈ N I , noting that J X j = G i j q ∗ j = J X j = G i j ¯ q j ( g k ) = ( g k ) i = G ik ,we have w = X j k G i j ¯ q j ( w ) + G ik ¯ q k ( w ) = X j k G i j ¯ q j ( w ) + Ã J X j = G i j q ∗ j ! ¯ q k ( w ) = G ˜ q and consequently C ( w ) = X j k h j ¯ q j ( w ) + h k ¯ q k ( w ) > X j k h j ¯ q j ( w ) + ¯ q k ( w ) J X j = h j q ∗ j = h · ˜ q ≥ C ( w )where the inequality in the above display is from (6.1) and from the fact that, by assumption,¯ q k ( w ) >
0. Thus we have a contradiction and therefore ¯ q k ( w ) = k ∈ S p which completesthe proof. (cid:3) Hereafter we fix a viable ranking ρ . As was noted in Theorem 6.1, there exists a continuousselection of the minimizer in (3.1). We now show that using the ranking ρ , one can give a ratherexplicit representation for such a selection function.Given w ∈ R I + , define q ∗ ( w ) ∈ R J + as follows. Set q ∗ j ( w ) = j ∈ S p . Define, q ∗ ρ (1) ( w ) = min i ∈ N ρ (1) { w i } µ ρ (1) . (6.2)For k ∈ {2, . . . , m } define, recursively, q ∗ ρ ( k ) ( w ) = min i ∈ N ρ ( k ) ( w i − k − X l = G i , ρ ( l ) q ∗ ρ ( l ) ( w ) ) µ ρ ( k ) . (6.3) Finally, for j ∈ S define q ∗ j ( w ) = ( w ˆ i ( j ) − m X k = G ˆ i ( j ), ρ ( k ) q ∗ ρ ( k ) ( w ) ) µ j , (6.4)where recall that ˆ i ( j ) is the unique resource processing the job j . By a recursive argument it is easyto check that q ∗ ( w ) defined above is a non-negative vector in R J . The following theorem showsthat q ∗ defined above is a continuous selection of the minimizer in (3.1). Theorem 6.3.
For any w ∈ R I + , q ∗ ( w ) ∈ Q s ( w ) and C ( w ) = h · q ∗ ( w ) = m X k = h ρ ( k ) q ∗ ρ ( k ) ( w ) + X j ∈ S h j q ∗ j ( w ). (6.5) Proof.
Fix w ∈ R I + . Let ¯ q ( w ) be as in Theorem 6.1. Then C ( w ) = h · ¯ q ( w ) and the proof of Theorem6.2 shows that ¯ q ( w ) ∈ Q s ( w ). Define s = sup © q ρ (1) : q ∈ Q s ( w ) and h · q = C ( w ) ª .Clearly the supremum is achieved, namely there is a ˇ q ∈ Q s ( w ) s.t. h · ˇ q = C ( w ) and ˇ q ρ (1) = s . Wenow show that s = q ∗ ρ (1) ( w ). First note that s ≤ q ∗ ρ (1) since from (6.2) there is an i ∗ ∈ N ρ (1) suchthat q ∗ ρ (1) ( w ) = w i ∗ µ ρ (1) = ( G ˇ q ) i ∗ µ ρ (1) ≥ ˇ q ρ (1) = s ,where the second equality holds since ˇ q ∈ Q s ( w ) and the next inequality is a consequence of thefact that i ∗ ∈ N ρ (1) . We now show that in fact the inequality can be replaced by equality. We argueby contradiction and suppose that s < q ∗ ρ (1) ( w ). For all i ∈ N ρ (1) define j ∗ ( i ) = arg max j ρ (1): i ∈ N j ½ ˇ q j µ j ¾ (6.6)and note that for any i ∈ N ρ (1) ˇ q j ∗ ( i ) µ j ∗ ( i ) ≥ J − Ã X j : i ∈ N j ˇ q j µ j − ˇ q ρ (1) µ ρ (1) ! > J µ w i − ˇ q ρ (1) µ ρ (1) ¶ ≥ J Ã q ∗ ρ (1) ( w ) − ˇ q ρ (1) µ ρ (1) ! ,where the second inequality uses the fact that ˇ q ∈ Q s ( w ) while the third uses (6.2) once more. Thus,min i ∈ N ρ (1) ½ ˇ q j ∗ ( i ) µ j ∗ ( i ) ¾ > q ∗ ρ (1) ( w ) − s J µ ρ (1) . (6.7)We can choose a subset M ∈ M S s , ρ (1) such that M ⊂ © j ∗ ( i ) : i ∈ N ρ (1) ª . From the definition of M , P j ∈ M K j − K ρ (1) is a nonnegative vector. Since ρ (1) ∈ O S m , due to Definition 4.2 µ ρ (1) h ρ (1) + C ( X j ∈ M K j − K ρ (1) ) ≤ C ( X j ∈ M K j ). (6.8)Thus there exists v ∈ Q s ¡P j ∈ M K j − K ρ (1) ¢ i.e., X j ∈ M K j − K ρ (1) = G v = J X j = K j b j where v j = b j µ j for j ∈ N J , (6.9)such that J X j = h j b j µ j = h · v = C ( X j ∈ M K j − K ρ (1) ). (6.10) ONTROL POLICIES FOR HGI PERFORMANCE 15
Furthermore, b ρ (1) =
0, since if b ρ (1) > P j ∈ M K i , j − K i , ρ (1) ≥ i ∈ N ρ (1) , so that for any l ∈ M we have P j ∈ M \{ l } K j − K ρ (1) ≥ M is not minimal and contradicts M ∈ M S s , ρ (1) .From (6.8) and (6.10) we have h ρ (1) µ ρ (1) + J X j = h j b j µ j ≤ X j ∈ M h j µ j . (6.11)Let u . = min j ∈ M ½ ˇ q j µ j ¾ (6.12)Since M ⊂ © j ∗ ( i ) : i ∈ N ρ (1) ª , from (6.7) u ≥ q ∗ ρ (1) ( w ) − s J µ ρ (1) . Define ˜ q ∈ R J + by˜ q ρ (1) = ˇ q ρ (1) + u µ ρ (1) , and ˜ q j = ˇ q j − { j ∈ M } u µ j + u b j µ j for j ρ (1). (6.13)By definition of u , ˜ q ∈ R J + . Also, w = J X j = K j µ ˇ q j µ j − { j ∈ M } u ¶ + u X j ∈ M K j = J X j = K j µ ˇ q j µ j − { j ∈ M } u ¶ + u J X j = K j b j + u K ρ (1) = J X j = K j ˜ q j µ j ,where the second equality uses (6.9) and last equality uses the observation that b ρ (1) =
0. Thus˜ q ∈ Q s ( w ). Furthermore, C ( w ) = J X j = h j ¡ ˇ q j − { j ∈ M } u µ j ¢ + u X j ∈ M h j µ j ≥ J X j = h j ¡ ˇ q j − { j ∈ M } u µ j ¢ + u h ρ (1) µ ρ (1) + u J X j = h j b j µ j = J X j = h j ˜ q j ≥ C ( w ),where the second line is from (6.11) and the last inequality holds since ˜ q ∈ Q s ( w ). So h · ˜ q = C ( w )and by definition of s , ˜ q ρ (1) ≤ s . However, since by assumption s < q ∗ ρ (1) ( w ),˜ q ρ (1) = s + u µ ρ (1) ≥ s + q ∗ ρ (1) ( w ) − s J > s (6.14)which is a contradiction. Thus we have shown s = q ∗ ρ (1) ( w ).Denote ˇ q as q . Then q ρ (1) = q ∗ ρ (1) ( w ). Note that C ( w ) = h · q = h ρ (1) q ∗ ρ (1) + X i ρ (1) h i q i .Let w = w − q ∗ ρ (1) ( w ) µ ρ (1) K ρ (1) . Then w = G h q − q ∗ ρ (1) ( w ) e ρ (1) i and if for any ˜ q ∈ R J + , G ˜ q = w , we have G h ˜ q + q ∗ ρ (1) ( w ) e ρ (1) i = G q = w and so h · ( ˜ q + q ∗ ρ (1) ( w ) e ρ (1) ) ≥ C ( w ) = h ρ (1) q ∗ ρ (1) ( w ) + X i ρ (1) h i q i . Thus h · ˜ q ≥ P i ρ (1) h i q i and since ˜ q is arbitrary vector in R J + satisfying G ˜ q = w C ( w ) = h · q − h ρ (1) q ∗ ρ (1) ( w ) = C ( w ) − h ρ (1) q ∗ ρ (1) ( w ).We now proceed via induction. Suppose that for some k ∈ {2, . . . , m } and all w ∈ R I + C ( w ) = k − X l = h ρ ( l ) q ∗ ρ ( l ) ( w ) + C ³ w k − ´ (6.15)where w k − = w − k − X l = q ∗ ρ ( l ) ( w ) µ ρ ( l ) K ρ ( l ) .Note that we have shown (6.15) for k =
2. With ¯ q as in Theorem 6.1 ¯ q ¡ w k − ¢ ∈ Q s ¡ w k − ¢ and C ³ w k − ´ = ¯ q ³ w k − ´ · h .Define s k = sup n q ρ ( k ) : q ∈ Q s ³ w k − ´ , q · h = C ( w k − ) o .Then there is ˇ q ∈ Q s ( w k − ) such that ˇ q ρ ( k ) = s k , and ˇ q · h = C ( w k − ). Also, using (6.3) we have forevery l < k an i ∗ ∈ N ρ ( l ) such that q ∗ ρ ( l ) ( w ) µ ρ ( l ) = w l − i ∗ . Thus,0 ≤ w k − i ∗ ≤ w i ∗ − l X u = G i ∗ , ρ ( u ) q ∗ ρ ( u ) ( w ) = w l − i ∗ − q ∗ ρ ( l ) ( w ) µ ρ ( l ) = l ∈
1, . . . k − i ∈ N ρ ( l ) such that w k − i =
0. (6.16)Since G ˇ q = w k − , this in turn says that ˇ q ρ ( l ) = l ∈
0, 1, . . . k −
1. Next, as for the case k =
1, wecan show that s k = q ∗ ρ ( k ) ( w ). Indeed, the inequality s k ≤ q ∗ ρ ( k ) follows on noting from (6.3) that forsome i ∗ ∈ N ρ ( k ) q ∗ ρ ( k ) ( w ) = w k − i ∗ µ ρ ( k ) = ( G ˇ q ) i ∗ µ ρ ( k ) ≥ ˇ q ρ ( k ) = s k .Next suppose s k < q ∗ ρ ( k ) ( w ). Define j ∗ ( i ) as in (6.6) replacing ρ (1) with ρ ( k ), then as before (using(6.3) instead of (6.2)) min i ∈ N ρ ( k ) ½ ˇ q j ∗ ( i ) µ j ∗ ( i ) ¾ > q ∗ ρ ( k ) ( w ) − s k J µ ρ ( k ) . (6.17)Thus from (6.17) we have that j ∗ ( i ) ∉ { ρ (1), . . . ρ ( k )}. We next claim that the set of resourcesassociated with ρ ( l ) for any l < k is not a subset of the set of resources associated with { j ∗ ( i ) : i ∈ N ρ ( k ) }. Indeed, if that were the case for some l < k , then we will have X i ∈ N ρ ( k ) K j ∗ ( i ) − K ρ ( l ) ≥
0. (6.18)From (6.16) there is an i ∗ such that K i ∗ , ρ ( l ) = w k − i ∗ =
0. Then from (6.18) K i ∗ , j ∗ ( i ) = i ∈ N ρ ( k ) . Since from (6.17) ˇ q j ∗ ( i ) >
0, we have w k − i ∗ > N ρ ( l ) S i ∈ N ρ ( k ) N j ∗ ( i ) for l =
1, . . . , k − M k ∈ M S s \ F k , ρ ( k ) F k such that M k ⊂ © j ∗ ( i ) : i ∈ N ρ ( k ) ª . ONTROL POLICIES FOR HGI PERFORMANCE 17
Since by definition ρ ( k ) ∈ O E k F k and by our choice M k ∈ M S s \ F k , ρ ( k ) F k , we have from Definition 4.2that there exists b k ∈ R J + such that b k ρ ( k ) = K ρ ( k ) + J X j = K j b kj = X j ∈ M k K j , and h ρ ( k ) µ ρ ( k ) + J X j = h j b kj µ j ≤ X j ∈ M k h j µ j .With u k as defined in (6.12) with M replaced by M k (and with ˇ q as above) u k ≥ q ∗ ρ ( k ) ( w ) − s k J µ ρ ( k ) .Define ˜ q as in (6.13) replacing ρ (1) with ρ ( k ), u with u k , and M with M k . Then as before h · ˜ q = C ( w k − ) and G ˜ q = w k − ; and as in the proof of (6.14) we see using (6.17) that ˜ q ρ ( k ) > s k whichcontradicts the definition of s k . This completes the proof that s k = q ∗ ρ ( k ) ( w ).Setting q k = ˇ q we have that q k ρ ( k ) = q ∗ ρ ( k ) ( w ). Also, recalling that w k = w k − − q ∗ ρ ( k ) ( w ) µ ρ ( k ) K ρ ( k ) and since G q k = w k − , we have G [ q k − q ∗ ρ ( k ) ( w ) e ρ ( k ) ] = w k and h · ( q k − q ∗ ρ ( k ) ( w ) e ρ ( k ) ) = C ( w k − ) − q ∗ ρ ( k ) ( w ) h ρ ( k ) . Furthermore, using the fact that h · q k = C ( w k − ), we have that if for ˜ q ∈ R J + , G ˜ q = w k , then h · ˜ q ≥ C ( w k − ) − q ∗ ρ ( k ) ( w ) h ρ ( k ) . Thus we have that C ( w k ) = C ( w k − ) − q ∗ ρ ( k ) ( w ) h ρ ( k ) .Combining this with the induction hypothesis (6.15), we have that (6.15) holds with k − k . This completes the induction step and proves (6.15) for all k =
2, . . . m +
1, in particular C ( w ) = m X l = h ρ ( l ) q ∗ ρ ( l ) ( w ) + C ¡ w m ¢ (6.19)where w m = w − m X l = K ρ ( l ) q ∗ ρ ( l ) ( w ). (6.20)Next, using (6.16) with k − m we see that for any q ∈ Q s ( w m ), q ρ ( l ) = l =
1, . . . m . Namely, C ( w m ) = X j ∈ S h j µ j w m ˆ i ( j ) .From the definition of w m in (6.20) and the definition of q ∗ j ( w ) for j ∈ S in (6.4) we then havethat C ( w m ) = X j ∈ S h j q ∗ j ( w ), w = m X l = K ρ ( l ) q ∗ ρ ( l ) ( w ) + X j ∈ S K j q ∗ j ( w ).This proves (6.5) and the statement that q ∗ ( w ) ∈ Q s ( w ), and completes the proof of the theorem. (cid:3) Analogous to ζ ki introduced in Section 4.1, let ζ i = { j ∈ S p : K i , j =
1} (6.21)be the set of primary jobs which impact node i . Theorem 6.4.
There exists B ∈ (0, ∞ ) such that for any q ∈ R J + and the corresponding workload,w = G q, we have ¯¯ h · q − C ( w ) ¯¯ ≤ B m X k = min i ∈ N ρ ( k ) X j ∈ ζ ki q j + I X i = X j ∈ ζ i q j .Proof. Recall from Theorem 6.3 that with q ∗ = q ∗ ( w ) C ( w ) = q ∗ · h = m X k = h ρ ( k ) q ∗ ρ ( k ) + X j ∈ S h j q ∗ j .Since q ∗ ρ (1) µ ρ (1) = min i ∈ N ρ (1) { w i } = min i ∈ N ρ (1) X j ∈ ζ i q j µ j + q ρ (1) µ ρ (1) we have q ρ (1) = q ∗ ρ (1) − min i ∈ N ρ (1) X j ∈ ζ i q j µ j µ ρ (1) from which we have 1 µ ρ (1) ¯¯¯ q ∗ ρ (1) − q ρ (1) ¯¯¯ ≤ min i ∈ N ρ (1) X j ∈ ζ i q j µ j .In general, for 2 ≤ k ≤ m we have q ∗ ρ ( k ) µ ρ ( k ) = min i ∈ N ρ ( k ) ( w i − k − X l = K i , ρ ( l ) q ∗ ρ ( l ) µ ρ ( l ) ) = min i ∈ N ρ ( k ) X j ∈ ζ ki q j µ j − k − X l = K i , ρ ( l ) ( q ∗ ρ ( l ) − q ρ ( l ) ) µ ρ ( l ) + q ρ ( k ) µ ρ ( k ) which gives 1 µ ρ ( k ) ¯¯¯ q ∗ ρ ( k ) − q ρ ( k ) ¯¯¯ ≤ min i ∈ N ρ ( k ) X j ∈ ζ ki q j µ j + k − X l = ¯¯¯ q ∗ ρ ( l ) − q ρ ( l ) ¯¯¯ µ ρ ( l ) .Consequently for k ∈ {2, ..., m } we have1 µ ρ ( k ) ¯¯¯ q ∗ ρ ( k ) − q ρ ( k ) ¯¯¯ ≤ min i ∈ N ρ ( k ) X j ∈ ζ ki q j µ j + k − X l = l min i ∈ N ρ ( k − − l ) X j ∈ ζ k − − li q j µ j .For j ∈ S we have with i = ˆ i ( j ) q ∗ j µ j = w i − m X k = K i , ρ ( k ) q ∗ ρ ( k ) µ ρ ( k ) = X j ′ ∈ ζ i q j ′ µ j ′ − m X k = K i , ρ ( k ) ( q ∗ ρ ( k ) − q ρ ( k ) ) µ ρ ( k ) + q j µ j which gives 1 µ j ¯¯¯ q ∗ j − q j ¯¯¯ ≤ X j ′ ∈ ζ i q j ′ µ ′ j + m X l = ¯¯¯ q ∗ ρ ( l ) − q ρ ( l ) ¯¯¯ µ ρ ( l ) ONTROL POLICIES FOR HGI PERFORMANCE 19
This, combined with our bounds on ¯¯¯ q ∗ ρ ( k ) − q ρ ( k ) ¯¯¯ for k ∈ {1, ..., m }, gives the following bound for j ∈ S ¯¯¯ q ∗ j − q j ¯¯¯ µ j ≤ X j ′ ∈ ζ i ( j ) q j ′ µ j ′ + m − X l = l min i ∈ N ρ ( m − l ) X j ′ ∈ ζ m − li q j ′ µ j ′ .Finally, for j ∈ S p we have ¯¯¯ q ∗ j − q j ¯¯¯ µ j = q j µ j ≤ min i ∈ N j X j ′ ∈ ζ i q j ′ µ j ′ .Combining the above bounds h · q = h · q ∗ + h · ( q − q ∗ ) ≤ C ( w ) + X j ∈ N J h j | q j − q ∗ |≤ C ( w ) + max j { h j } X j ¯¯¯ q j − q ∗ j ¯¯¯ ≤ C ( w ) + max j { h j } max j { µ j } J J m X k = min i ∈ N ρ ( k ) X j ∈ ζ ki q j µ j + I X i = X j ∈ ζ i q j µ j ≤ C ( w ) + max j { h j } max j { µ j }min j { µ j } J J m X k = min i ∈ N ρ ( k ) X j ∈ ζ ki q j + I X i = X j ∈ ζ i q j .Because h · q ≥ C ( w ) we have ¯¯ h · q − C ( w ) ¯¯ ≤ B m X k = min i ∈ N ρ ( k ) X j ∈ ζ ki q j + I X i = X j ∈ ζ i q j where B = max j { h j }max j { µ j }min j { µ j } J J . (cid:3)
7. S
OME P ROPERTIES OF THE R ATE A LLOCATION P OLICY
In this section we record some important properties of the rate allocation policy x ( · ) introducedin Definition 4.5. Throughout this section y ( t ), x ( t ) and E rj ( t ) will be as in Definition 4.5. Ourfirst result shows that x satisfies basic conditions for admissibility, namely, it is nonnegative andsatisfies the capacity constraint. Lemma 7.1.
For all t ≥ , x ( t ) ≥ and K x ( t ) ≤ C .Proof.
For the first statement in the lemma it suffices to show that y j ( t ) ≥ j ∈ N J and t ≥ δ it is clear that y j ( t ) ≥ j ∈ N J \ S and for j ∈ S with ˆ i ( j ) ∉ ̟ r ( t ).Consider now a j ∈ S for which ˆ i ( j ) ∈ ̟ r ( t ). Then y j ( t ) = C ˆ i ( j ) − X l j : K ˆ i ( j ), l = y l ( t ).Also note that X l j : K ˆ i ( j ), l = y l ( t ) ≤ X l j : K ˆ i ( j ), l = ¡ ̺ l + δ ¢ ≤ X l j : K ˆ i ( j ), l = ̺ l + min j ′ { ̺ j ′ }2 and thus since K ̺ = Cy j ( t ) = C ˆ i ( j ) − X l j : K ˆ i ( j ), l = y l ( t ) ≥ C ˆ i ( j ) − X l j : K ˆ i ( j ), l = ̺ l − min j ′ { ̺ j ′ }2 ≥ ̺ j − min j ′ { ̺ j ′ }2 ≥ K x ( t ) ≤ C forall t ≥
0. Let i ∈ N I be arbitrary. It suffices to show that for all t ≥ C i ≥ P Jj = K i , j y j ( t ). Fromdefinition of y j ( t ) for j ∈ S in Definition 4.5, it is clear that when i ∈ ̟ r ( t ), C i = P Jj = K i , j y j ( t ).Finally, if i ∉ ̟ r ( t ), then Definition 4.5 gives y j ( t ) < ̺ j for all j with K i , j = J X j = K i , j y j ( t ) < J X j = K i , j ̺ j < C i .This completes the proof. (cid:3) The following two results are used in the proof of Theorem 8.1.
Lemma 7.2.
For all t ≥ and i ∈ ̟ r ( t ) such that P Jj = K i , j E rj ( t ) = we have C i = P Jj = K i , j x j ( t ) .Proof. Let t ≥ i ∈ ̟ r ( t ) satisfy P Jj = K i , j E rj ( t ) =
0. Then for all j with K i , j = x j ( t ) = y j ( t ) and so it suffices to prove that C i = P Jj = K i , j y j ( t ). However, this is an immediate consequenceof the definition of y j ( t ) for j ∈ S and ˆ i ( j ) ∈ ̟ r ( t ) in Definition 4.5. (cid:3) From Condition 2.2 we can find ˆ R ∈ (0, ∞ ) such that for all r ≥ ˆ R and j ∈ N J we have ¯¯¯ ̺ j − ̺ rj ¯¯¯ ≤ − m − δ J , 2 λ j ≥ λ rj ≥ λ j /2, and 2 µ j ≥ µ rj ≥ µ j /2. (7.1)For the rest of this work we will assume without loss of generality that r ≥ ˆ R . Lemma 7.3.
For all t ≥ and j ∈ N J if c r α ≤ Q rj ( t ) < c r α then λ rj − µ rj x j ( t ) ≥ µ j − m − δ J .Proof.
Note that if E rj ( t ) = x j ( t ) = r ≥ ˆ R , implies on recalling the definition of δ from Definition 4.5 that λ rj − µ rj x j ( t ) = λ rj ≥ λ j /2 = µ j ̺ j /2 ≥ µ j δ .Thus the result holds in this case.We now consider the case E rj ( t ) = x j ( t ) = y j ( t ). If j ∈ N J \ S or j ∈ S and ˆ i ( j ) ∉ ̟ r ( t ),Definition 4.5 gives y j ( t ) ≤ ̺ j − − m − δ J which combined with (7.1) implies λ rj − µ rj x j ( t ) ≥ λ rj − µ rj µ ̺ j − − m − δ J ¶ = µ rj ³ ̺ rj − ̺ j ´ + µ rj − m − δ J ≥ − µ rj − m − δ J + µ j − m − δ J ≥ µ j − m − δ J and the result again holds. Finally we consider the remaining case, namely j ∈ S , E rj ( t ) = i ( j ) ∈ ̟ r ( t ). We will consider two sub-cases. ONTROL POLICIES FOR HGI PERFORMANCE 21
Case 1: ζ i ( j ) ∩ σ r ( t )
6= ; . Let l ∗ ∈ ζ i ( j ) ∩ σ r ( t ). Then y l ∗ ( t ) = ̺ l ∗ + δ and y j ( t ) = C ˆ i ( j ) − X l j : K ˆ i ( j ), l = y l ( t ) = C ˆ i ( j ) − y l ∗ ( t ) − m X k = K ˆ i ( j ), ρ ( k ) y ρ ( k ) ( t ) − X l ∈ ζ i ( j ) : l l ∗ y l ( t ).Furthermore, − m X k = K ˆ i ( j ), ρ ( k ) y ρ ( k ) ( t ) ≤ − m X k = K ˆ i ( j ), ρ ( k ) ³ ̺ ρ ( k ) − k − m − δ ´ ≤ − m X k = K ˆ i ( j ), ρ ( k ) ̺ ρ ( k ) + δ ¡ − − m − ¢ and − X l ∈ ζ i ( j ) : l l ∗ y l ( t ) ≤ − X l ∈ ζ i ( j ) : l l ∗ µ ̺ l − − m − δ J ¶ ≤ − X l ∈ ζ i ( j ) : l l ∗ ̺ l + − m − δ .Consequently y j ( t ) ≤ C ˆ i ( j ) − X l j : K ˆ i ( j ), l = ̺ l − δ + δ ¡ − − m − ¢ + − m − δ ≤ ̺ j − δ − m − which combined with (7.1) gives λ rj − µ rj y j ( t ) ≥ λ rj − µ rj ¡ ̺ j − − m − δ ¢ ≥ µ rj ³ ̺ rj − ̺ j ´ + µ rj − m − δ ≥ − µ rj − m − δ J + µ j − m − δ ≥ − µ j − m − δ J + µ j − m − δ ≥ µ j − m − δ and the result holds. Case 2: ζ i ( j ) ∩ σ r ( t ) = ; . In this case the assumption ˆ i ( j ) ∈ ̟ r ( t ) implies that there exists some k ∈ N m such that K ˆ i ( j ), ρ ( k ) = ρ ( k ) ∈ σ r ( t ). Let k ∗ = max{ k ∈ N m : K ˆ i ( j ), ρ ( k ) = ρ ( k ) ∈ σ r ( t )}.Consequently ζ k ∗ ˆ i ( j ) ∩ σ r ( t ) = ; and ρ ( k ∗ ) ∈ σ r ( t ) so y ρ ( k ∗ ) = ̺ ρ ( k ∗ ) + k ∗ − m − δ . Recall that y j ( t ) = C ˆ i ( j ) − X l j : K ˆ i ( j ), l = y l ( t ) = C ˆ i ( j ) − y ρ ( k ∗ ) − k ∗ − X k = K ˆ i ( j ), ρ ( k ) y ρ ( k ) ( t ) − m X k = k ∗ + K ˆ i ( j ), ρ ( k ) y ρ ( k ) ( t ) − X l ∈ ζ i ( j ) K ˆ i ( j ), l y l ( t ).For the third term on the right side, we have − k ∗ − X k = K ˆ i ( j ), ρ ( k ) y ρ ( k ) ( t ) ≤ − k ∗ − X k = K ˆ i ( j ), ρ ( k ) ³ ̺ ρ ( k ) − k − m − δ ´ ≤ − k ∗ − X k = K ˆ i ( j ), ρ ( k ) ̺ ρ ( k ) + ³ − − k ∗ + ´ k ∗ − m − δ ≤ − k ∗ − X k = K ˆ i ( j ), ρ ( k ) ̺ ρ ( k ) + k ∗ − m − δ − − m − δ . By the definition of k ∗ for all k ∈ { k ∗ +
1, ..., m } if K ˆ i ( j ), ρ ( k ) = ρ ( k ) ∉ σ r ( t ) and ζ k ˆ i ( j ) ∩ σ r ( t ) =; , consequently y ρ ( k ) ( t ) = ̺ ρ ( k ) − − k − m − δ . This gives − m X k = k ∗ + K ˆ i ( j ), ρ ( k ) y ρ ( k ) ( t ) = − m X k = k ∗ + K ˆ i ( j ), ρ ( k ) ³ ̺ ρ ( k ) − − k − m − δ ´ ≤ − m X k = k ∗ + K ˆ i ( j ), ρ ( k ) ̺ ρ ( k ) + − k ∗ − m − δ (1 − − m + k ∗ ).Finally, by assumption, ζ i ( j ) ∩ σ r ( t ) = ; and therefore − X l ∈ ζ i ( j ) K ˆ i ( j ), l y l ( t ) = − X l ∈ ζ i ( j ) K ˆ i ( j ), l µ ̺ l − − m − δ J ¶ ≤ − X l ∈ ζ i ( j ) K ˆ i ( j ), l ̺ l + − m − δ .This gives y j ( t ) ≤ C ˆ i ( j ) − ³P l j : K ˆ i ( j ), l = ̺ l ´ − k ∗ − m − δ + k ∗ − m − δ − − m − δ + − k ∗ − m − δ + − m − δ ≤ ̺ j − − m − δ which combined with (7.1) implies λ rj − µ rj y j ( t ) ≥ λ rj − µ rj ¡ ̺ j − − m − δ ¢ = µ rj ³ ̺ rj − ̺ j ´ + µ rj − m − δ ≥ − µ rj − m − δ J + µ j − m − δ ≥ − µ j − m − δ J + µ j − m − δ ≥ µ j − m − δ and completes the proof. (cid:3) The following lemma will be used in the proofs of Propositions 8.3 and 8.4.
Lemma 7.4. (a) Let t ≥ and k ∈ N m be such that ζ ki ′ ∩ σ r ( t )
6= ; for all i ′ ∈ N ρ ( k ) . Then for anyi ∈ N ρ ( k ) satisfying P j ∈ ζ ki E rj ( t ) = , we have X j ∈ ζ ki ³ ̺ rj − x j ( t ) ´ ≤ − − m − δ . (7.2) (b) Let i ∈ N I and t ≥ be such that ζ i ∩ σ r ( t )
6= ; and P j ∈ ζ i E rj ( t ) = . Then, we have X j ∈ ζ i ³ ̺ rj − x j ( t ) ´ ≤ − − δ .Proof. (a) Recall that we assume r ≥ ˆ R and consequently (7.1) holds. Let k ∈ N m and t ≥ ζ ki ′ ∩ σ r ( t )
6= ; for all i ′ ∈ N ρ ( k ) . Let i ∈ N ρ ( k ) be such that P j ∈ ζ ki E rj ( t ) =
0. We need to show that(7.2) holds for such an i . Since ζ ki ′ ∩ σ r ( t )
6= ; for all i ′ ∈ N ρ ( k ) , Definition 4.5 gives y ρ ( k ) ( t ) = ̺ ρ ( k ) − k − m − δ . (7.3)Since P j ∈ ζ ki E rj ( t ) =
0, for all j ∈ ζ ki , x j ( t ) = y j ( t ) so to prove (7.2) it suffices to show X j ∈ ζ ki ³ ̺ rj − y j ( t ) ´ ≤ − − m − δ . (7.4)Due to the assumption that ζ ki ∩ σ r ( t )
6= ; we have i ∈ ̟ r ( t ) and consequently Definition 4.5 gives y ˇ j ( i ) ( t ) = C i − X j ˇ j ( i ): K i , j = y j ( t ). ONTROL POLICIES FOR HGI PERFORMANCE 23
Therefore X j ∈ ζ ki y j ( t ) = y ˇ j ( i ) ( t ) + X j ∈ ζ ki : j ˇ j ( i ) y j ( t ) = C i − X j ˇ j ( i ): K i , j = y j ( t ) + X j ∈ ζ ki : j ˇ j ( i ) y j ( t ) = C i − y ρ ( k ) ( t ) − k − X v = K i , v y ρ ( v ) ( t ).However, from (7.3) and Definition 4.5 C i − y ρ ( k ) ( t ) − k − X v = K i , v y ρ ( v ) ( t ) ≥ C i − ³ ̺ ρ ( k ) − k − m − δ ´ − k − X v = K i , v ¡ ̺ ρ ( v ) + v − m − δ ¢ ≥ C i − k X v = K i , v ̺ ρ ( v ) + k − m − δ − k − m − δ + − m − δ ≥ X j ∈ ζ ki ̺ j + − m − δ which gives X j ∈ ζ ki y j ( t ) ≥ X j ∈ ζ ki ̺ j + − m − δ .Combining this with (7.1) gives X j ∈ ζ ki ³ ̺ rj − y j ( t ) ´ = X j ∈ ζ ki ̺ rj − X j ∈ ζ ki y j ( t ) ≤ X j ∈ ζ ki ³ ̺ rj − ̺ j ´ − − m − δ ≤ J − m − δ J − − m − δ ≤ − − m − δ .This proves (7.4) and completes the proof of part (a).(b) Suppose now that i ∈ N I and t ≥ ζ i ∩ σ r ( t )
6= ; and P j ∈ ζ i E rj ( t ) =
0. From thelatter property we have x j ( t ) = y j ( t ) for all j ∈ ζ i , and because ζ i ∩ σ r ( t )
6= ; there exists l ∗ ∈ ζ i such that l ∗ ∈ σ r ( t ). From Definition 4.5 y l ∗ ( t ) = ̺ l ∗ + δ and X j ∈ ζ i y j ( t ) = y l ∗ ( t ) + X j ∈ ζ i : j l ∗ y j ( t ) ≥ ̺ l ∗ + δ + X j ∈ ζ i : j l ∗ µ ̺ j − − m − δ J ¶ ≥ X j ∈ ζ i ̺ j + δ − − m − δ ≥ X j ∈ ζ i ̺ j + δ X j ∈ ζ i ³ ̺ rj − x j ( t ) ´ = X j ∈ ζ i ̺ rj − X j ∈ ζ i y j ( t ) ≤ X j ∈ ζ i ̺ rj − X j ∈ ζ i ̺ j − δ ≤ X j ∈ ζ i ³ ̺ rj − ̺ j ´ − δ ≤ J − m − δ J − δ ≤ − − δ .This completes the proof of (b). (cid:3)
8. L
ARGE D EVIATION E STIMATES
Recall the allocation scheme x ( · ) given by Definition 4.5 and define processes Q r , B r , T r asso-ciated with this allocation scheme with ˙ B r ( t ) = x ( t ), t ≥
0, as in Section 2. Also recall the otherassociated processes as defined in (2.4) –(2.6). Note that the allocation scheme depends on a pa-rameter α ∈ (0, 1/2) and c , c ∈ (0, ∞ ). Let X r ( t ) = ( Q r ( t ), E r ( t )) and letˆ X r ( t ) = ¡ ˆ Q r ( t ), E r ( r t ) ¢ = ¡ Q r ( r t )/ r , E r ( r t ) ¢ , t ≥
0. (8.1)Note that although ˆ Q r is not Markovian, the pair ˆ X r defines a strong Markov process with statespace S r . = ( R + ∩ r N ) J × {0, 1} J . Expectations of various functionals of the Markov process ˆ X r whenˆ X r (0) = x will be denoted as E x and the associated probabilities by P x . The following theorem is akey step in estimating the idleness terms in state dynamics. Theorem 8.1.
For any ǫ ∈ (0, ∞ ) and j ∈ N J there exist ˆ B , ˆ B , ˆ B , ˆ B , R ∈ (0, ∞ ) such that for all r ≥ R,t ≥ and x ∈ S r we haveP x ÃZ tr I { E rj ( s ) = d s ≥ ǫ r + α /2 t ! ≤ ˆ B e − r + α /2 t ˆ B + µ + ˆ B r + α /2 ¶ − ˆ B r t (8.2) and P x ÃZ tr I { E rj ( s ) = d s ≥ ǫ r t ! ≤ ˆ B e − r t ˆ B + µ + ˆ B r + α ¶ − ˆ B r t . (8.3) Proof.
Let j ∈ N J , x ∈ S r and ǫ > c , c from Section 4.1. Define τ r , j . = inf n s ≥ Q rj ( s ) ≥ c r α o , τ r , j l − . = inf n s ≥ τ r , j l − : Q rj ( s ) < r α c + c o ,and τ r , j l . = inf n s ≥ τ r l − : Q rj ( s ) ≥ c r α o for all l ≥
1. Recall the functions E j introduced in Definition 4.5. Define the indicator functions θ r , jl . =
1, if E rj ( s ) = s ∈ ³ τ r , j l − , τ r , j l i
0, otherwise.For t > η r , jt = max n l : τ r , j l − ≤ t r o , ˆ η r , jt = max n l : τ r , j l − ≤ t r o (8.4)and N r , jk = P kl = θ r , jl . Consider the events, B r , j = n η r , jt ≤ λ rj r t o , ˆ B r , j = n ˆ η r , jt ≤ λ rj r t o , B r , j = ( N r , j l λ rj tr m ≤ λ rj ǫ c − c ) r − α /2 t ) , ˆ B r , j = ( N r , j l λ rj tr m ≤ λ rj ǫ c − c ) r − α t ) .Let C r . = (Z r t I n E rj ( s ) = o d s ≥ ǫ r + α /2 t ) , ˆ C r . = (Z r t I n E rj ( s ) = o d s ≥ ǫ r t ) .Then P ¡ C r ¢ ≤ P ³ ( B r , j ) c ´ + P ³ ( B r , j ) c ´ + P ³ B r , j ∩ B r , j ∩ C r ´ (8.5) ONTROL POLICIES FOR HGI PERFORMANCE 25 and P ¡ ˆ C r ¢ ≤ P ³ ( ˆ B r , j ) c ´ + P ³ ( ˆ B r , j ) c ´ + P ³ ˆ B r , j ∩ ˆ B r , j ∩ ˆ C r ´ . (8.6)Noting that each occurrence of τ r , j l − requires an arrival of a job of type j , we have P ³ ( B r , j ) c ´ = P ³ η r , jt > λ rj r t ´ ≤ P ³ A rj ( t r ) ≥ λ rj r t ´ .Similarly, P ³ ( ˆ B r , j ) c ´ ≤ P ³ A rj ( t r ) ≥ λ rj r t ´ .Thus from the first inequality in Theorem A.1 in Appendix we can find R ∈ (0, ∞ ) and κ , κ ∈ (0, ∞ )such that for all r ≥ R , t ≥ j ∈ N J P ³ ( B r , j ) c ´ ≤ κ e − tr κ , P ³ ( ˆ B r , j ) c ´ ≤ κ e − tr κ . (8.7)We now estimate P ³ ( B r , j ) c ´ , P ³ ( ˆ B r , j ) c ´ . Note that the n θ r , jl o ∞ l = are i.i.d. Bernoulli with parameter p ( r ) where p ( r ) = P ( θ r , jl = = P ³ Q rj ( ς r , jl ) < c r α ´ and ς r , jl . = inf n s ≥ τ r , j l − : Q rj ( t ) < c r α or Q rj ( t ) ≥ c r α o . (8.8)The probability p ( r ) can be estimated as follows. Note that from Lemma 7.3, for τ r , j l − ≤ s < ς r , jl λ rj − µ rj x j ( s ) ≥ µ j κ (8.9)where κ . = − m − δ J . Letting ¯ C = max i { C i } and d j . = ( c − c )/( µ j κ ), define A r , jl = ½ sup ≤ s ≤ d j r α ¯¯¯ A rj ( τ r , j l − + s ) − A rj ( τ r , j l − ) − λ rj s ¯¯¯ + sup ≤ s ≤ ¯ Cd j r α ¯¯¯ S rj ( B rj ( τ r , j l − ) + s ) − S rj ( B rj ( τ r , j l − )) − µ rj s ¯¯¯ ≥ ( c − c ) r α ¾ .From Theorem A.1 and strong Markov property there exist κ , κ ∈ (0, ∞ ) and R ∈ [ R , ∞ ) such thatfor all r ≥ R , j ∈ N J , and l ≥ P ³ A r , jl ´ ≤ κ e − r α κ .We can also assume without loss of generality that for r ≥ R , r α c − c >
2. From (8.9), on the event ³ A r , jl ´ c , we have for s ∈ h τ r , j l − , ς r , jl ∧ ³ τ r , j l − + d j r α ´´ Q rj ( s ) ≥ r α c + c − + ³ A rj ( s ) − A rj ( τ r , j l − ) ´ − ³ S rj ( B rj ( s )) − S rj ( B rj ( τ r , j l − )) ´ ≥ r α c + c − − r α c − c + ( s − τ r , j l − ) µ j ∆ .Since the expression on the right side with s = τ r , j l − + d j r α is larger than c r α we have that on ³ A r , jl ´ c , ς r , jl < τ r , j l − + d j r α and so Q rj ( ς r , jl ) > c r α . Thus ³ A r , jl ´ c ∩ { θ r , jl = = ; and p ( r ) ≤ P ³ A r , jl ´ ≤ κ e − r α κ .Choose R ∈ [ R , ∞ ) such that for all r ≥ R we have ǫ /[10( c − c ) r + α ] ≥ p ( r ), ǫ /[5( c − c ) r + α /2 ] ≤ ³ λ rj r + ´ /5 ≤ λ rj r /2. (8.10) so in particular from the third inequality, for all t ≥ l λ rj t r m ¡ ǫ /[5( c − c ) r + α /2 ] ¢ ≤ λ rj t r − α /2 ǫ /[2( c − c ]) (8.11)and l λ rj t r m ¡ ǫ /[5( c − c ) r + α ] ¢ ≤ λ rj t r − α ǫ /[2( c − c )]. (8.12)Note that if Z ∼ Bin( L , p ) then, for all u > P ( Z ≥ u ) ≤ (1 + p ( e − L e − u .Thus we have P Ã N r l λ rj tr m ≥ λ rj ǫ c − c ) r − α /2 t ! ≤ e − λ rj ǫ c − c r − α /2 t ¡ + p ( r )( e − ¢ l λ rj tr m ≤ µ + p ( r ) e ǫ /5( c − c ) r + α /2 ¶ l λ rj tr m .where the second line uses (8.11) and the fact that if for positive a , b , c , d , ab ≤ c , then e − c (1 + d ( e − b ≤ µ + de a ¶ b .For all r ≥ R we have µ + p ( r ) e ǫ /[5( c − c ) r + α /2 ] ¶ l λ rj tr m ≤ µ + ǫ /[10( c − c ) r + α /2 ]1 + ǫ /[5( c − c ) r + α /2 ] ¶ l λ rj tr m ≤ µ + ǫ /[50( c − c ) r + α /2 ] ¶ l λ rj tr m ≤ µ + ǫ /[50( c − c )] r + α /2 ¶ − λ j r t (8.13)where the first line uses the inequality e x ≥ + x and the first bound in (8.10), the second uses thesecond bound in (8.10) along with the inequality (1 + x )/(1 + x ) ≤ + x ) for x ∈ [0, 1/4], and thethird uses (7.1) to bound λ rj by λ j . Thus we have shown P ³ ( B r , j ) c ´ ≤ µ + ˆ B r + α /2 ¶ − ˆ B r t (8.14)where ˆ B = ǫ /[50( c − c )] and ˆ B =
1. A similar calculation shows that P ³ ( ˆ B r , j ) c ´ ≤ µ + ˆ B r + α ¶ − ˆ B r t (8.15)Finally we estimate the third probability on the right sides of (8.5) and (8.6). Note that Z tr I n E rj ( s ) = o d s ≤ Z τ r , j I n E rj ( s ) = o d s + η r , jt X l = Z τ r , j l τ r , j l − I n E rj ( s ) = o d s From (8.8) we see that Z τ r , j l τ r , j l − I { E j ( s ) = } d s = τ r , j l − ς r , jl . (8.16) ONTROL POLICIES FOR HGI PERFORMANCE 27
Indeed, if θ r , jl = ς r , jl = τ r , j l and the integral on the left side is 0. Also, if θ r , jl = Q rj ( ς r , jl ) =⌈ c r α ⌉ − ς r , jl < τ r , j l and E j ( s ) = s ∈ [ ς r , jl , τ r , j l ], giving once more the identity in (8.16). Inthe latter case we also have the representation τ r , j l − ς r , jl = inf n s ≥ A rj ( ς r , jl + s ) − A rj ( ς r , jl ) ≥ § c r α ¨ − § c r α ¨ + o . (8.17)Similarly if we define ς r , j = inf n s ≥ Q rj ( t ) ≤ c r α or Q rj ( t ) ≥ c r α o then Z τ r , j I n E rj ( s ) = o d s = τ r , j − ς r , j where if ς r , j < τ r , j we have τ r , j − ς r , j = inf n s ≥ A rj ( ς r , j + s ) − A rj ( ς r , j ) ≥ § c r α ¨ − § c r α ¨ + o . (8.18)Consequently , since on B r , j , η r , jt ≤ λ rj r t , by taking r suitably large P ³ B r , j ∩ B r , j ∩ C r ´ ≤ P B r , j ∩ τ r , j − ς r , j + l λ rj tr m X l = ³ τ r , j l − ς r , jl ´ ≥ ǫ r + α /2 t ≤ P à inf ( s ≥ A rj ( s ) ≥ à λ rj ǫ r + α /2 t !) ≥ ǫ r + α /2 t ! ≤ P à ˇ A rj ( ǫ r + α /2 t ) ≤ λ rj ǫ r + α /2 t ! where ˇ A rj is a Poisson process with rate λ rj and the second inequality comes from the representa-tions in (8.18) and (8.17). From Theorem A.1 there exist κ , κ ∈ (0, ∞ ) and R ∈ [ R , ∞ ) such thatfor all r ≥ R P ³ B r , j ∩ B r , j ∩ C r ´ ≤ P µ sup ≤ s ≤ ǫ r + α /2 t ¯¯¯ ˇ A rj ( s ) − λ r s ¯¯¯ > ǫ r + α /2 t ¶ ≤ κ e − r + α /2 t κ . (8.19)A similar calculation shows that P ³ ˆ B r , j ∩ ˆ B r , j ∩ ˆ C r ´ ≤ κ e − r t κ . (8.20)Finally (8.7), (8.14), (8.19), and (8.5) prove (8.2) while (8.7), (8.15), (8.20) and (8.6) prove (8.3).This completes the proof. (cid:3) Let c . = Jc min j µ j and recall that ¯ C . = max i ∈ N I { C i }. Note that if for given s ≥ W ri ( s ) > c r α forsome i ∈ N I then we must have that Q rj ( s ) ≥ c r α for some j ∈ N J with K i j =
1, namely i ∈ ˆ ω r ( s ).From Lemma 7.2 it then follows that for such a s if C i > P Jj = K i j x j ( s ), the E rj ( t ) j with K i j =
1. From this it follows that for any t ≥ Z t I { W ri ( s ) > c r α } ( s ) d I ri ( s ) ≤ ¯ C X j : K i j = Z t I { E rj ( s ) = d s . This along with Theorem 8.1 implies that for any ǫ > i ∈ N I there exist ˆ B , ˆ B , ˆ B , ˆ B , R ∈ (0, ∞ )such that for all r ≥ R , t ≥ x ∈ S r we have P x ÃZ tr I { W ri ( s ) ≥ c r α } ( s ) d I ri ( s ) ≥ ǫ r + α /2 t ! ≤ ˆ B e − r + α /2 t ˆ B + µ + ˆ B r + α /2 ¶ − ˆ B r t (8.21)and P x ÃZ tr I { W ri ( s ) ≥ c r α } ( s ) d I ri ( s ) ≥ ǫ r t ! ≤ ˆ B e − r t ˆ B + µ + ˆ B r + α ¶ − ˆ B r t . (8.22)8.1. Estimating holding cost through workload cost.
Recall the matrix M introduced in Section3. Along with the process ˆ W r = K M r ˆ Q r , it will be convenient to also consider the process ˜ W r . = K M ˆ Q r . The following is the main result of the section which says that under the scheme intro-duced in Definition 4.5, the queue lengths for the associated workload are ‘asymptotically optimal’in a certain sense. This result will be key in showing that under our policy, property (II) of HGIholds asymptotically. Theorem 8.2.
There exist B , R ∈ (0, ∞ ) such that for all r ≥ R, x = ( q , z ) ∈ S r , θ > and T ≥ , wehave ¯¯¯¯ E x ·Z ∞ e − θ t h · ˆ Q r ( t ) d t ¸ − E x ·Z ∞ e − θ t C ¡ ˜ W r ( t ) ¢ d t ¸¯¯¯¯ ≤ Br α − + | q | − e − θ and ¯¯¯¯ E x · T Z T h · ˆ Q r ( t ) d t ¸ − E x · T Z T C ¡ ˜ W r ( t ) ¢ d t ¸¯¯¯¯ ≤ Br α − (1 + | q | ) . In order to prove the result we begin with the following two propositions.Recall the sets ζ i , ζ ki from (6.21) and (4.3) and that c = Jc min j µ j . For ξ ≥ i ∈ N I and k ∈ N m letˆ τ i ( ξ ) . = inf t ≥ ξ : X j ∈ ζ i Q rj ( s ) µ rj < c r α , ˆ τ sk ( ξ ) . = inf t ≥ ξ : min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( s ) µ rj < c r α . (8.23) Proposition 8.3.
There exist R , B ∈ (0, ∞ ) such that for all r ≥ R, i ∈ N I , x = ( q , z ) ∈ S r , and k ∈ N m we have r E x Z ˆ τ i (0)0 X j ∈ ζ i Q rj ( s ) µ j d s ≤ B (1 + | q | ) r − and r E x Z ˆ τ sk (0)0 min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( s ) µ j d s ≤ B (1 + | q | ) r − . Proof.
Let k ∈ N m be arbitrary. Note that under P x , Q r (0) = r ˆ Q r (0) = r q . Choose ˇ i (0) ∈ N ρ ( k ) suchthat X j ∈ ζ k ˇ i (0) r q j µ j = min i ∈ N ρ ( k ) X j ∈ ζ ki r q j µ j and define d = X j ∈ ζ k ˇ i (0) q j µ j and ∆ = − m − δ , (8.24) ONTROL POLICIES FOR HGI PERFORMANCE 29 where δ is as in Definition 4.5. If r d < c r α then ˆ τ sk (0) = r d ≥ c r α so that ˆ τ sk (0) >
0. We claim that for t ∈ £
0, ˆ τ sk (0) ¢ and i ′ ∈ N ρ ( k ) we have ζ ki ′ ∩ σ r ( t ) . To see the claim note that for such t , for all i ′ ∈ N ρ ( k ) , from the definition of ˆ τ sk (0) X j ∈ ζ ki ′ Q rj ( t ) µ rj ≥ min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( t ) µ rj ≥ c r α .Thus, from the definition of c there is a j ∈ ζ ki ′ such that Q rj ( t ) ≥ c J r α µ rj ≥ c J r α µ j ≥ c r α ,namely j ∈ σ r ( t ). Thus we have ζ ki ′ ∩ σ r ( t )
6= ; proving the claim. From Lemma 7.4(a) we now havethat for i ∈ N ρ ( k ) and t ∈ £
0, ˆ τ sk (0) ¢ such that P j ∈ ζ ki E rj ( t ) = X j ∈ ζ ki ³ ̺ rj − x j ( t ) ´ ≤ − − m − δ = − ∆ . (8.25)Recall that ¯ C = max i { C i }. Define for y ≥
0, the events A ry = ( X i ∈ N I Z (2 r y / ∆ ) ∧ ˆ τ sk (0)0 ½P j ∈ ζ ki E rj ( s ) > ¾ d s ≥ yr
4( ¯ C ∨ ∆ ) ) and B ry = [ j ∈ N J ( sup ≤ t ≤ r y / ∆ ¯¯¯ A rj ( t ) − t λ rj ¯¯¯ + sup ≤ t ≤ Cr y / ∆ ¯¯¯ S rj ( t ) − t µ rj ¯¯¯ ≥ y ¯ µ min r J ) .From Theorem 8.1 (cf. (8.2) with r y ∆ substituted in for t ) and Theorem A.1 there exist B , B ∈ (0, ∞ ) and R ∈ £ ˆ R , ∞ ¢ (recall ((7.1))) such that for all r ≥ R and y ≥ max{ ∆ , d , 1}, P ³ A ry [ B ry ´ ≤ B e − B y .Also on the event ³ A ry S B ry ´ c for all t ∈ £
0, ˆ τ sk (0) ∧ r y / ∆ ¢ we havemin i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( t ) µ rj ≤ X j ∈ ζ k ˇ i (0) Q rj ( t ) µ rj ≤ r d + X j ∈ ζ k ˇ i (0) A rj ( t ) µ rj − X j ∈ ζ k ˇ i (0) S rj ( B j ( t )) µ rj ≤ r d + X j ∈ ζ k ˇ i (0) y ¯ µ min r J µ rj + X j ∈ ζ k ˇ i (0) ³ t ̺ rj − B j ( t ) ´ where the last line follows from the definition of the event B ry . Next note that B j ( t ) = Z t x j ( s ) d s = Z t x j ( s ) ½P j ∈ ζ k ˇ i (0) E rj ( s ) = ¾ d s + Z t x j ( s ) ½P j ∈ ζ k ˇ i (0) E rj ( s ) > ¾ d s .From (8.25) , on the above event, for t ∈ £
0, ˆ τ sk (0) ∧ r y / ∆ ¢X j ∈ ζ k ˇ i (0) Z t x j ( s ) ½P j ∈ ζ k ˇ i (0) E rj ( s ) = ¾ d s ≥ Z t ( X j ∈ ζ k ˇ i (0) ̺ rj ) ½P j ∈ ζ k ˇ i (0) E rj ( s ) = ¾ d s + ∆ Z t ½P j ∈ ζ k ˇ i (0) E rj ( s ) = ¾ d s Thus, recalling the definition of A ry X j ∈ ζ k ˇ i (0) ³ t ̺ rj − B j ( t ) ´ ≤ Z t X j ∈ ζ k ˇ i (0) ( ̺ rj − x j ( s )) ½P j ∈ ζ k ˇ i (0) E rj ( s ) ¾ d s − ∆ t + ∆ Z t ½P j ∈ ζ k ˇ i (0) E rj ( s ) ¾ d s ≤ C ˇ i (0) yr C − ∆ t + ∆ yr ∆ ≤ r y − ∆ t and consequently on the event ³ A ry S B ry ´ c for all t ∈ £
0, ˆ τ sk (0) ∧ r y / ∆ ¢ we have (since y ≥ d )min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( t ) µ rj ≤ r ( d + y ) − ∆ t ≤ r y − ∆ t .Since at t = r y / ∆ , 2 r y − ∆ t =
0, we must have ˆ τ sk (0) < r y / ∆ so that on the above event Z ˆ τ sk (0)0 min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( t ) µ rj d t ≤ ∆ r y .This gives for r ≥ R and y ≥ max{ d , 1} P x Z ˆ τ sk (0)0 min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( t ) µ rj d t > ∆ r y ≤ B e − B y .A straightforward calculation now shows that E x Z ˆ τ sk (0)0 min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( t ) µ rj d t ≤ r B (1 + | q | )where B depends only on B , B and δ . This proves the second statement in the lemma. The proofof the first statement follows in a very similar manner and is omitted. (cid:3) The following proposition will be the second ingredient in the proof of Theorem 8.2.
Proposition 8.4.
There exist H , R ∈ (0, ∞ ) such that for all r ≥ R, i ∈ N I , k ∈ N m , and ≤ T < T < ∞ satisfying T − T ≥ we have r E Z ˆ τ i ( r T )ˆ τ i ( r T ) X j ∈ ζ i Q rj ( s ) µ rj d s ≤ ( T − T ) H r α − and r E Z ˆ τ sk ( r T )ˆ τ sk ( r T ) min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( s ) µ rj d s ≤ ( T − T ) H r α − Proof.
Once again we only prove the second statement since the proof of the first statement issimilar. Many steps in the proof are similar to those in Proposition 8.3 but we give details to keepthe proof self contained. Let k ∈ N m be arbitrary. Recall ¯ µ min = min j ∈ N J { µ j }. and ¯ C = max i ∈ N I { C i }.Also let for k ∈ N m Z rk ( t ) . = min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( t ) µ rj . (8.26) ONTROL POLICIES FOR HGI PERFORMANCE 31
Define the stopping times, τ . = r T and for l ∈ N τ l − . = inf © t ≥ τ l − : Z rk ( t ) ≥ c r α ª , τ l = inf © t ≥ τ l − : Z rk ( t ) < c r α ª .Let ˆ l . = min{ l ≥ τ l + > r T }. Then recalling the definition of ˆ τ sk ( ξ ) from (8.23), ˆ τ sk ( r T ) = r T ∨ τ l . Consequently we can write E "Z ˆ τ sk ( r T )ˆ τ sk ( r T ) Z rk ( s ) d s ≤ E "Z τ ∧ r T ˆ τ sk ( r T ) Z rk ( s ) d s + E " ∞ X l = I { τ l ≤ r T } Z τ l + ∧ r T τ l Z rk ( s ) d s + E " ∞ X l = I { τ l + ≤ r T } Z τ l + τ l + Z rk ( s ) d s . (8.27)By definition, for all s ∈ £ ˆ τ sk ( r T ), τ ∧ r T ¢ and s ∈ £ τ l , τ l + ∧ r T ¢ we have Z rk ( s ) ≤ c r α whichgives E "Z τ ∧ r T ˆ τ sk ( r T ) Z rk ( s ) d s + E " ∞ X l = I { τ l ≤ r T } Z τ l + ∧ r T τ l Z rk ( s ) d s ≤ c r α + ( T − T ). (8.28)For all l ∈ N let ˇ i ( l ) ∈ N ρ ( k ) satisfy X j ∈ ζ k ˇ i ( l ) Q rj ( τ l + ) µ rj = min i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( τ l + ) µ rj = Z rk ( τ l + )and note that X j ∈ ζ k ˇ i ( l ) Q rj ( τ l + ) µ rj ≤ c r α + µ min .Recall the definition of ∆ in (8.24) and define for y ∈ R + and l ∈ N , the events A rl , y = ( X i ∈ N I Z ( τ l + + r + α /2 y / ∆ ) ∧ τ l + τ l + I ½P j ∈ ζ ki E rj ( s ) > ¾ d s ≥ r + α /2 ¡ ¯ C ∨ ∆ ¢ y ) and B rl , y = ( X j ∈ N J sup τ l + ≤ t ≤ τ l + + r + α /2 y / ∆ ¯¯¯ A rj ( t ) − A rj ( τ l + ) − ( t − τ l + ) λ rj ¯¯¯ + X j ∈ N J sup τ l + ≤ t ≤ τ l + + ¯ C r + α /2 y / ∆ ¯¯¯ S rj ( t ) − S rj ( τ l + ) − ( t − τ l + ) µ rj ¯¯¯ ≥ ¯ µ min r + α /2 y ) From the strong Markov property, Theorems 8.1 (cf. (8.1)) and A.1 there exist B , B ∈ (0, ∞ ) and R ∈ £ ˆ R , ∞ ¢ such that for all r ≥ R , y ≥ ∆ /2, and l ∈ N we have r + α /2 ∆ /2 > µ min and P ³ A rl , y [ B rl , y ´ ≤ B e − B y . (8.29)We claim that for t ∈ [ τ l + , τ l + ) we have ζ ki ′ ∩ σ r ( t )
6= ; for all i ′ ∈ N ρ ( k ) . To see the claim note thatfor all i ′ ∈ N ρ ( k ) , P j ∈ ζ ki ′ Q rj ( t ) µ rj ≥ min i ∈ N ρ ( k ) P j ∈ ζ ki Q rj ( t ) µ rj ≥ c r α . Thus, from the definition of c there isa j ∈ ζ ki ′ such that Q rj ( t ) ≥ c J r α µ rj ≥ c J r α µ j ≥ c r α , namely j ∈ σ r ( t ). Thus we have ζ ki ′ ∩ σ r ( t )
6= ; proving the claim. From Lemma 7.4(a) for i ∈ N ρ ( k ) and t ∈ [ τ l + , τ l + ) such that P j ∈ ζ ki E rj ( t ) = X j ∈ ζ ki ³ ̺ rj − x j ( t ) ´ ≤ − − m − δ = − ∆ . (8.30)Consequently on the event ³ A rl , y S B rl , y ´ c for all t ∈ £ τ l + , τ l + ∧ ¡ τ l + + r + α /2 y / ∆ ¢¢ we havemin i ∈ N ρ ( k ) X j ∈ ζ ki Q rj ( t ) µ rj ≤ X j ∈ ζ k ˇ i ( l ) Q rj ( τ l + ) µ rj + X j ∈ ζ k ˇ i ( l ) Q rj ( t ) µ rj − X j ∈ ζ k ˇ i ( l ) Q rj ( τ l + ) µ rj ≤ c r α + µ min + X j ∈ ζ k ˇ i ( l ) µ rj h³ A rj ( t ) − A rj ( τ l + ) ´ + ³ S rj ( B j ( t )) − S rj ( B j ( τ l + )) ´i ≤ c r α + µ min + r + α /2 y + X j ∈ ζ k ˇ i ( l ) ³ ( t − τ l + ) ̺ rj − ( B j ( t ) − B j ( τ l + )) ´ where the last line comes from the definition of the event B rl , y . Note that for all j ∈ N J and t ≥ τ l + we have B j ( t ) − B j ( τ l + ) = Z t τ l + x j ( s ) d s = Z t τ l + x j ( s ) I ½P j ∈ ζ ki E rj ( s ) > ¾ d s + Z t τ l + x j ( s ) I ½P j ∈ ζ ki E rj ( s ) = ¾ d s .From (8.30), on the above event and for t ∈ £ τ l + , τ l + ∧ ¡ τ l + + r + α /2 y / ∆ ¢¤Z t τ l + X j ∈ ζ ki x j ( s ) I ½P j ∈ ζ ki E rj ( s ) = ¾ d s ≥ Z t τ l + X j ∈ ζ ki ̺ rj I ½P j ∈ ζ ki E rj ( s ) = ¾ d s + ∆ Z t τ l + I ½P j ∈ ζ ki E rj ( s ) = ¾ d s (8.31)so that X j ∈ ζ k ˇ i ( l ) ³ ( t − τ l + ) ̺ rj − ( B j ( t ) − B j ( τ l + )) ´ ≤ Z t τ l + X j ∈ ζ k ˇ i ( l ) ³ ̺ rj − x j ( s ) ´ I ½P j ∈ ζ ki E rj ( s ) > ¾ d s − ∆ ( t − τ l + ) + ∆ Z t τ l + I ½P j ∈ ζ ki E rj ( s ) > ¾ d s ≤ C ˇ i ( l ) r + α /2 C y − ∆ ( t − τ l + ) + r + α /2 y ≤ r + α /2 y − ∆ ( t − τ l + )where the second line follows because we are on the set ³ A rl , y ´ c . Consequently on the event ³ A rl , y S B rl , y ´ c for t ∈ £ τ l + , τ l + ∧ ¡ τ l + + r + α /2 y / ∆ ¢¢ Z rk ( t ) ≤ c r α + µ min + r + α /2 y − ∆ ( t − τ l + ). (8.32)The right side of (8.32) with t = τ l + + r + α /2 y / ∆ equals2 c r α + µ min + r + α /2 y − ∆ (2 r + α /2 y / ∆ ) < c r α ONTROL POLICIES FOR HGI PERFORMANCE 33 where the inequality is from (8.29), and so we must have τ l + < τ l + + r + α /2 y / ∆ . This com-bined with (8.32) gives on the event ³ A rl , y S B rl , y ´ c Z τ l + τ l + Z rk ( s ) d s ≤ K y r + α for a K < ∞ depending only on c , ¯ µ min and ∆ . Then for y ≥ B = max{2 c + µ min , ∆ } we have from(8.29) P X r ( τ l + ) µZ τ l + τ l + Z rk ( t ) d t > K y r + α ¶ ≤ B e − B y and a standard argument now gives E X r ( τ l + ) ·Z τ l + τ l + Z rk ( t ) d t ¸ ≤ B r + α (8.33)where the constant B depends only on B , B , B and K . Let L r = max{ l ≥ τ l + ≤ T r }.Note that for all l ≥ τ l + implies an arrival of a job of type j ∈ S i ∈ N ρ ( k ) ζ ki in theinterval ( τ l , τ l + ], so that for some K ∈ (0, ∞ ) E x L r ≤ K r ( T − T ) for all x ∈ S r Consequently E " ∞ X l = I { τ l + ≤ r T } Z τ l + τ l + Z rk ( s ) d s ≤ B r + α E x £ L r ¤ ≤ B r + + α ( T − T ),where B . = K B . This, combined with (8.27) and (8.28) gives1 r E "Z ˆ τ sk ( r T )ˆ τ sk ( r T ) Z rk ( s ) d s ≤ ¡ c r α − + B r α − ¢ ( T − T ).The result follows. (cid:3) We can now complete the proof of Theorem 8.2.
Proof of Theorem 8.2.
Let R < ∞ be given by the maximum of the two R values from Proposi-tions 8.3 and 8.4. Note that by (3.1) , for all t ≥ h · ˆ Q r ( t ) ≥ C ¡ ˜ W r ( t ) ¢ (8.34)and by Theorem 6.4, there is a B ∈ (0, ∞ ) such that for all t , r , h · ˆ Q r ( t ) − C ¡ ˜ W r ( t ) ¢ ≤ B X k ∈ N m min i ∈ N ρ ( k ) X j ∈ ζ ki ˆ Q rj ( t ) µ rj + I X i = X j ∈ ζ i ˆ Q rj ( t ) µ rj . (8.35)Let Z rk be as in (8.26). From monotone convergence we have for θ ≥ n →∞ r E "Z ˆ τ sk ( r n )0 e − θ t / r Z rk ( t ) d t = E Z ∞ e − θ t min i ∈ N ρ ( k ) X j ∈ ζ ki ˆ Q rj ( t ) µ rj d t . (8.36) Note that 1 r E "Z ˆ τ sk ( r n )0 e − θ t / r Z rk ( t ) d t = r E ·Z ˆ τ sk (0)0 e − θ t / r Z rk ( t ) d t ¸ + n X l = r E "Z ˆ τ sk ( r l )ˆ τ sk ( r ( l − e − θ t / r Z rk ( t ) d t .From Proposition 8.3, we have for some B ∈ (0, ∞ ), for r ≥ R , θ ≥ x ∈ S r ,1 r E x ·Z ˆ τ sk (0)0 e − θ t / r Z rk ( t ) d t ¸ ≤ B r − (1 + | q | ). (8.37)Also, from Theorem 8.4, there is B ∈ (0, ∞ ) such that for k ∈ N m , r ≥ R and any l ∈ N r E "Z ˆ τ sk ( r l )ˆ τ sk ( r ( l − e − θ t / r Z rk ( t ) d t ≤ r e − θ ( l − E "Z ˆ τ sk ( r l )ˆ τ sk ( r ( l − Z rk ( t ) d t ≤ B e − θ ( l − r α − Consequently for r ≥ R r E "Z ˆ τ sk ( r n )0 e − θ t / r Z rk ( t ) d t ≤ Ã B (1 + | q | ) + B n − X l = e − l θ ! r α − .Sending n → ∞ , using (8.36), we have for θ > k ∈ N m E Z ∞ e − θ t min i ∈ N ρ ( k ) X j ∈ ζ ki ˆ Q rj ( t ) µ rj d t ≤ µ B (1 + | q | ) + B − e − θ ¶ r α − .A similar argument shows that there are B , B ∈ (0, ∞ ) such that for all i ∈ N I and r ≥ RE Z ∞ e − θ t X j ∈ ζ i ˆ Q rj ( t ) µ rj d t ≤ µ B (1 + | q | ) + B − e − θ ¶ r α − .Combining the above two estimates with (8.34) and (8.35) we have the first inequality in the theo-rem.For the second inequality, we write E x T Z T min i ∈ N ρ ( k ) X j ∈ ζ ki ˆ Q rj ( t ) µ rj d t = E x " Tr Z Tr Z rk ( t ) d t ≤ r E x · T Z ˆ τ sk (0)0 Z rk ( t ) d t ¸ + r E x " T Z ˆ τ sk ( r T )ˆ τ sk (0) Z rk ( t ) d t . Applying (8.37) with θ = T ≥ r E x · T Z ˆ τ sk (0)0 Z rk ( t ) d t ¸ ≤ B r − (1 + | q | ). (8.38)Also, from Theorem 8.4 for r ≥ R we have, for some ˜ B ∈ (0, ∞ ) and all T ≥ k ∈ N m ,1 r E x " T Z ˆ τ sk ( r T )ˆ τ sk (0) Z rk ( t ) d t ≤ T ˜ B r α − T ≤ ˜ B r α − . ONTROL POLICIES FOR HGI PERFORMANCE 35
Consequently, for all T ≥ k ∈ N m E x T Z T min i ∈ N ρ ( k ) X j ∈ ζ ki ˆ Q rj ( t ) µ rj d t ≤ B r − (1 + | q | ) + ˜ B r α − A similar argument shows that for some ˜ B , ˜ B ∈ (0, ∞ ), and all i ∈ N I , T ≥ E x T Z T X j ∈ ζ i ˆ Q rj ( t ) µ rj d t ≤ ˜ B r − (1 + | q | ) + ˜ B r α − .Combining the above two estimates with (8.34) and (8.35) once more, we have the second inequal-ity in the theorem. (cid:3) Lyapunov function and uniform moment estimates.
In this section we establish uniform in t and r moment bounds on ˆ W r ( t ). The following is the main result of this section. Theorem 8.5.
There exist β , γ , R , H ∈ (0, ∞ ) such that for all i ∈ N I , t ≥ and r ≥ RE x h e γ ˆ W ri ( t ) i ≤ H ³ + e − β t V i ( x ) ´ .The proof is given at the end of the section. Letˇ τ ri , ξ . = inf © t ≥ ξ : ¯¯ ˆ W ri ( t ) ¯¯ ≤ c ª , (8.39)where recall that c = Jc min j µ j . We begin by establishing a bound on certain exponential momentsof ˇ τ ri , ξ . Proposition 8.6.
There exist δ ∗ , R ∈ (0, ∞ ) and H : R + → R + such that for all i ∈ N I , r ≥ R and < β < δ ∗ E x h e β ˇ τ ri , ξ i < H ( β ) e H ( β )( w i + ξ ) for all x = ( q , z ) ∈ S r and ξ ≥ , where w = G r q.Proof. Fix i ∈ N I . Given x = ¡ q , z ¢ ∈ S r let w i = ( G r q ) i . Recall the definition of v ∗ given in Condi-tion 2.2. Fix ξ ≥ t ≥ max{2 ξ , 8 w i / v ∗ i , 1} . = M ξ . (8.40)Consider the events A ri , t = (Z r t I © W ri ( t ) ≥ c r α ª ( s ) d I ri ( s ) ≥ v ∗ i C i r t ) and B ri , t = [ j ∈ N J ( sup ≤ s ≤ r t ¯¯¯ A rj ( s ) − s λ rj ¯¯¯ + sup ≤ s ≤ C i r t ¯¯¯ S rj ( s ) − s µ rj ¯¯¯ ≥ min{1, ¯ µ min } v ∗ i J r t ) .Using (8.22) and Theorem A.1 we can choose ˆ H , ˆ H ∈ (0, ∞ ) and R ∈ ( ˆ R , ∞ ) such that for all r ≥ R and t ≥ P x ³ A ri , t [ B ri , t ´ ≤ ˆ H e − t ˆ H ,where ˆ R was introduced above (7.1). Furthermore, we can assume that R is large enough so thatfor all r ≥ R , 2 v ∗ i r ≥ C i − J X j = K i , j ρ rj ≥ v ∗ i r , c r α − + µ min r − ≤ c . (8.41) Then, for all r ≥ R and s , s ∈ [0, r t ] satisfying s > s , on the event ³ A ri , t S B ri , t ´ c , we have J X j = K i , j µ rj S rj ( B rj ( s )) − J X j = K i , j µ rj S rj ( B rj ( s )) ≥ J X j = K i , j ³ B rj ( s ) − B rj ( s ) ´ − J X j = K i , j min{1, ¯ µ min } v ∗ i J µ rj r t ≥ J X j = K i , j ³ B rj ( s ) − B rj ( s ) ´ − v ∗ i r t and J X j = K i , j µ rj A rj ( s ) − J X j = K i , j µ rj A rj ( s ) ≤ J X j = K i , j ρ rj ( s − s ) + J X j = K i , j min{1, ¯ µ min } v ∗ i J µ rj r t ≤ J X j = K i , j ρ rj ( s − s ) + v ∗ i r t .Let σ = k ≥ σ k − = inf{ s ≥ σ k − : W ri ( t ) ≥ c r α }, σ k = inf{ s ≥ σ k − : W ri ( t ) < c r α }.Then, on the event ³ A ri , t S B ri , t ´ c , for any σ k − < ξ r , k ≥
1, we have on noting that W i ( σ k − ) ≤ c r α + µ min + w i r sup σ k − ≤ s ≤ σ k ∧ ξ r W ri ( s ) ≤ sup σ k − ≤ s ≤ σ k ∧ ξ r à W ri ( σ k − ) + J X j = K i , j µ rj A rj ( s ) − J X j = K i , j µ rj A rj ( σ k − ) − à J X j = K i , j µ rj S rj ( B rj ( s )) − J X j = K i , j µ rj S rj ( B rj ( σ k − )) !! ≤ sup σ k − ≤ s ≤ σ k ∧ ξ r à J X j = K i , j ρ rj ( s − σ k − ) − J X j = K i , j ³ B rj ( s ) − B rj ( σ k − ) ´ + c r α + µ min + w i r + v ∗ i r t ¶ ≤ c r α + µ min + w i r + v ∗ i r t where the third inequality follows from recalling that we are on the event ³ A ri , t ´ c so J X j = K i , j h ρ rj ( s − σ k − ) − ( B rj ( s ) − B rj ( σ k − ) i ≤ à J X j = K i , j ρ rj − C i ! ( s − σ k − ) + v ∗ i r t ≤ − v ∗ i r ( s − σ k − ) + v ∗ i r t ≤ v ∗ i r t .Thus on the event ³ A ri , t S B ri , t ´ c we haveˆ W ri ( ξ ) ≤ v ∗ i t + w i + c r α − + µ min r − . ONTROL POLICIES FOR HGI PERFORMANCE 37
Consequently on the event ³ A ri , t S B ri , t ´ c ∩ { ˇ τ ri , ξ > t } we have, by a similar calculation,ˆ W ri ( t ) = ˆ W ri ( ξ ) + ¡ ˆ W ri ( t ) − ˆ W ri ( ξ ) ¢ ≤ v ∗ i t + w i + c r α − + µ min r − + r ( t − ξ ) J X j = K i , j ρ rj − r C i ( t − ξ ) + v ∗ i t ≤ v ∗ i t + w i + c r α − + µ min r − − t à C i − J X j = K i , j ρ rj ! r ≤ v ∗ i t − t v ∗ i + w i + c r α − + µ min r − ≤ c ,where the third and the fourth inequalities follow from (8.41) and recalling that t ≥ max{2 ξ , 8 w i / v ∗ i }.Since on the set { ˇ τ ri , ξ > t } we must have ˆ W ri ( t ) > c we have arrived at a contradiction. Conse-quently ³ A ri , t S B ri , t ´ c ∩ { ˇ τ ri , ξ ( x ) > t } = ; and P x ³ ˇ τ ri , ξ > t ´ = P x ³³ A ri , t [ B ri , t ´ ∩ { ˇ τ ri , ξ > t } ´ ≤ P x ³ A ri , t [ B ri , t ´ ≤ ˆ H e − t ˆ H .Thus for β < ˆ H E x h e β ˇ τ ri , ξ i ≤ + β e β M ξ + β ˆ H − β e ( β − ˆ H ) M ξ ≤ H ( β ) e H ( β )( ξ + w i ) for suitable H ( β ) ∈ (0, ∞ ), where the last inequality comes from the definition of M ξ in (8.40). (cid:3) We now establish a lower bound on an exponential moment of ˇ τ ri ,0 . Proposition 8.7.
For all i ∈ N I there exist R , H , H , H ∈ (0, ∞ ) such that for all r ≥ R, β > andx = ( q , z ) ∈ S r satisfying w i = ( G r q ) i ≥ H we haveE x h e β ˇ τ ri ,0 i > H e H β w i Proof.
For k ∈ (0, ∞ ) define the event B ri , k = [ j ∈ N J ( sup ≤ s ≤ k ¯¯¯¯¯ A rj ( r s ) r − r s λ rj ¯¯¯¯¯ + sup ≤ s ≤ C i k ¯¯¯¯¯ S rj ( r s ) r − r s µ rj ¯¯¯¯¯ ≥ v ∗ i min{1, ¯ µ min } k J ) .From Theorem A.1 there exists R ∈ ( ˆ R , ∞ ) (recall (7.1)) and ˆ H , ˆ H ∈ (0, ∞ ) such that for all r ≥ R and k ∈ (0, ∞ ), we have P ³ B ri , k ´ ≤ ˆ H e − k ˆ H .We assume that R is big enough so that (8.41) is satisfied for all r ≥ R . Let H = max ½ c , v ∗ i log(2 ˆ H )ˆ H ¾ .Then for w i ≥ H we have P ³ B ri , w i /(6 v ∗ i ) ´ ≤ ˆ H e − w i ˆ H /6 v ∗ i ≤
12 (8.42) and on the event ( B ri , w i /6 v ∗ i ) c we have from (2.2) and (2.6), that under P x ,inf ≤ s ≤ w i /6 v ∗ i ˆ W ri ( s ) = inf ≤ s ≤ w i /6 v ∗ i à w i + J X j = K i , j A rj ( r s ) r µ rj − J X j = K i , j S rj ( r ¯ B rj ( s )) r µ rj ! ≥ inf ≤ s ≤ w i /6 v ∗ i à w i − r à C i − J X j = K i , j ̺ rj ! s − J X j = K i , j min{1, ¯ µ min } w i J µ rj ! ≥ inf ≤ s ≤ w i /6 v ∗ i µ w i − v ∗ i s ¶ ≥ w i > c where the third line uses (7.1) and (8.41). Thus { ˇ τ ri ,0 ≤ w i /6 v ∗ i } ∩ ( B ri , w i /6 v ∗ i ) c = ; , P x a.s.. This gives E x h e β ˇ τ ri ,0 i = E x · e β ˇ τ ri ,0 I B ri , wi /6 v ∗ i ¸ + E x · e β ˇ τ ri ,0 I ( B ri , wi /6 v ∗ i ) c ¸ ≥ e β ( w i /(6 v ∗ i ) ) P x ³ B ri , w i /6 v ∗ i ´ c ≥ e ( β /(6 v ∗ i )) w i ,where the last inequality is from (8.42). Thus completes the proof. (cid:3) Recall δ ∗ from Proposition 8.6 and fix β ∈ (0, δ ∗ ). For i ∈ N I let V i ( x ) . = E x h e β ˇ τ ri ,0 i .Also recall the Markov process ˆ X r in (8.1). The following result proves a Lyapunov function prop-erty for V i . Proposition 8.8.
There exist H , R ∈ (0, ∞ ) such that for all x = ( q , z ) ∈ S r , r ≥ R, i ∈ N I , and t ∈ [0, 1] we have E x £ V i ( ˆ X r ( t )) ¤ ≤ e − β t V i ( x ) + H .Proof.
From the Markov property we have E x £ V i ( ˆ X r ( t )) ¤ = E x · e β ³ ˇ τ ri , t − t ´ ¸ = E x · e β ³ ˇ τ ri , t − t ´ I n ˇ τ ri ,0 ≥ t o ¸ + E x · e β ³ ˇ τ ri , t − t ´ I n ˇ τ ri ,0 < t o ¸ .Let R be as in Proposition 8.6. Let t ∈ [0, 1] and r ≥ R be arbitrary. Then from Proposition 8.6, forsome ˆ H , ˆ H ∈ (0, ∞ ) E x · e β ³ ˇ τ ri , t − t ´ I n ˇ τ ri ,0 < t o ¸ ≤ sup x ′ : w i ≤ c sup ≤ ξ ≤ E x ′ h e β ˇ τ ri , ξ i ≤ ˆ H e ˆ H (2 c + Furthermore, E x · e β ³ ˇ τ ri , t − t ´ I n ˇ τ ri ,0 ≥ t o ¸ = e − t β E x · e β ˇ τ ri ,0 I n ˇ τ ri ,0 ≥ t o ¸ ≤ e − t β E x h e β ˇ τ ri ,0 i = e − t β V i ( x ).Combining the two estimates we have the result. (cid:3) From the Lyapunov function property proved in the previous result we have the following mo-ment estimate for all time instants.
Proposition 8.9.
There exist H , H , R ∈ (0, ∞ ) such that for all t ≥ , i ∈ N I and r ≥ R we haveE x £ V i ( ˆ X r ( t )) ¤ ≤ H e − β t V i ( x ) + H ONTROL POLICIES FOR HGI PERFORMANCE 39
Proof.
Let R , H be as in Proposition 8.8. Then for all i ∈ N I , x = ( q , z ) ∈ S r , t ∈ [0, 1] and r ≥ R , wehave E x £ V i ( ˆ X r ( t )) ¤ ≤ e − β t V i ( x ) + H .Then from the Markov property, for any r ≥ R and t ≥ E x £ V i ( ˆ X r ( t )) ¤ = E x £ E x £ V i ( ˆ X r ( t )) ¯¯ ˆ X r ( ⌊ t ⌋ ) ¤¤ ≤ H + e − β ( t −⌊ t ⌋ ) E x £ V i ( ˆ X r ( ⌊ t ⌋ )) ¤ . (8.43)Using the Markov property again E x £ V i ( ˆ X r ( ⌊ t ⌋ )) ¤ = E x £ E £ V i ( ˆ X r (1)) | ˆ X r ( ⌊ t ⌋ − ¤¤ ≤ H + e − β E x £ V i ( ˆ X r ( ⌊ t ⌋ − ¤ .Iterating the above inequality we get E x £ V i ( ˆ X r ( ⌊ t ⌋ )) ¤ ≤ e − β ⌊ t ⌋ V i ( x ) + H ⌊ t ⌋− X k = ³ e − β ´ k ≤ e − β ⌊ t ⌋ V i ( x ) + H − e − β .Combining this with (8.43) we have for all t ≥ E x £ V i ( ˆ X r ( t )) ¤ ≤ e − β t V i ( x ) + H µ + − e − β ¶ .The result follows. (cid:3) Proof of Theorem 8.5.
This proof is immediate from Proposition 8.7 and Proposition 8.9 ontaking γ = H β where H is as in the statement of Proposition 8.7 and β is as fixed above Proposition8.8. 9. P ATH O CCUPATION M EASURE C ONVERGENCE
Let for t ≥ Z r ( t ) = w r + G r ( ˆ A r ( t ) − ˆ S r ( ¯ B r ( t ))), (9.1)where w r = G r q . Consider the collection of random variables indexed by T and r taking values in P ¡ D ([0, 1] : R I + × R I ) ¢ , defined by θ rT ( d x × d y ) = T Z T δ ˆ W r ( t +· ) ( d x ) δ ˆ Z r ( t +· ) − ˆ Z r ( t ) ( d y ) d t .In this section we will prove the tightness of the collection { θ rT , T > r >
0} of random path occu-pation measures and characterize limit points along suitable subsequences.We begin by noting the following monotonicity property of a one dimensional Skorohod mapintroduced in Section 3.
Theorem 9.1.
Fix T ∈ (0, ∞ ) and f ∈ D ([0, T ] : R ) satisfying f (0) = . Let ϕ = Γ ( f ) . Suppose ϕ , ϕ ∈ D ([0, T ] : R ) are such that • ϕ ( t ) = f ( t ) + h ( t ) , t ∈ [0, T ] , where h ∈ D ([0, T ] : R ) is a nondecreasing function withh (0) = and R [0, T ] (0, ∞ ) ( ϕ ( s )) d h ( s ) = . • ϕ ( t ) = f ( t ) + h ( t ) , t ∈ [0, T ] , where h ∈ D ([0, T ] : R ) is a nondecreasing function withh (0) = and ϕ ( t ) ≥ for all t ∈ [0, T ] .Then for all t ∈ [0, T ] , ϕ ( t ) ≤ ϕ ( t ) ≤ ϕ ( t ) .Proof. The proof of the second inequality is straightforward and is omitted. Consider now thefirst inequality. Note that ϕ ( t ) = f ( t ) + h ( t ) where h ( t ) = − inf ≤ s ≤ t { f ( s )} and thus it suffices toshow that for any t ∈ [0, T ], h ( t ) ≤ − inf ≤ s ≤ t { f ( s )}. Assume that there exists t ∗ ∈ [0, T ] such that h ( t ∗ ) > − inf ≤ s ≤ t ∗ { f ( s )} . = a . Let t ∗ = sup{ s ∈ [0, t ∗ ] : h ( s ) ≤ a } and note that either h ( t ∗ ) > a or h ( t ∗ ) = a and h ( r ) > a for all r ∈ ( t ∗ , t ∗ ]. In the first case ϕ ( t ∗ ) = f ( t ∗ ) + h ( t ∗ ) > f ( t ∗ ) − inf ≤ s ≤ t ∗ { f ( s )} ≥ ϕ ( t ∗ ) > Z { t ∗ } d h ( s ) = h ( t ∗ ) − lim s ↑ t ∗ h ( s ) > r ∈ ( t ∗ , t ∗ ] ϕ ( r ) = f ( r ) + h ( r ) > f ( r ) − a ≥ f ( r ) − inf ≤ s ≤ t ∗ { f ( s )} ≥ ϕ ( r ) > r ∈ ( t ∗ , t ∗ ] and Z ( t ∗ , t ∗ ] d h ( s ) = h ( t ∗ ) − h ( t ∗ ) = h ( t ∗ ) − a > t ∈ [0, T ] we have h ( t ) ≤ − inf ≤ s ≤ t { f ( s )} and thedesired inequality follows. (cid:3) Theorem 9.2.
For any ǫ > and T ∈ (0, ∞ ) there exists R ∈ (0, ∞ ) such that for all r ≥ R and x = ( q , z ) ∈ S r , sup s ∈ [0, ∞ ) P x µ sup ≤ t ≤ T ¯¯ Γ ¡ ˆ W r ( s ) + ˆ Z r ( s + · ) − ˆ Z r ( s ) + r ( K ρ r − C ) ι ¢ ( t ) − ˆ W r ( t + s ) ¯¯ > ǫ ¶ < ǫ .Proof. We will only prove the result without the outside supremum and in fact only when s = x ∈ Γ r . Let ˆ ξ ri ( t ) = r Z r t I © W ri ( s ) ≥ c r α ª ( s ) d I ri ( s ), i ∈ N I .Note thatˆ W ri ( t ) − c r α − = ˆ Z ri ( t ) + t r ( K ρ r − C ) i + ˆ ξ ri ( t ) − c r α − + Z t I © ˆ W ri ( s ) − c r α − < ª ( s ) d ˆ I ri ( s )and consequently due to Theorem 9.1 we haveˆ W r ( t ) − c r α − ≤ Γ ¡ ˆ Z r + r ( K ρ r − C ) ι + ˆ ξ r − c r α − ¢ ( t ), t ≥ W r ( t ) = ˆ Z r ( t ) + t r ( K ρ r − C ) + ˆ I r ( t )is a nonnegative function and ˆ I r (0) is nondecreasing and satisfies ˆ I r (0) =
0. Thus once more fromTheorem 9.1 Γ ¡ ˆ Z r ( · ) + r ( K ρ r − C ) ι ¢ ≤ ˆ W r ( t ), t ≥ t ≥ Γ ¡ ˆ Z r + r ( K ρ r − C ) ι ¢ ( t ) ≤ ˆ W r ( t ) ≤ Γ ¡ ˆ Z r + r ( K ρ r − C ) ι + ˆ ξ r ( · ) − c r α − ¢ + c r α − . (9.2)Lipschitz property of the Skorokhod map gives that there is a κ ∈ (0, ∞ ) such that for all T > ≤ t ≤ T ¯¯ Γ ¡ ˆ Z r ( · ) + r ( K ρ r − C ) ι ¢ ( t ) − ˆ W r ( t ) ¯¯ ≤ κ ¡ c r α − + ¯¯ ˆ ξ r ( T ) ¯¯¢ .From Theorem 8.1 (see (8.22)), for any ǫ > T ∈ (0, ∞ ), there exists R ∈ (0, ∞ ) such that for all r ≥ R and x ∈ S r P x ¡¯¯ ˆ ξ r ( T ) ¯¯ > ǫ ¢ < ǫ .The result follows. (cid:3) ONTROL POLICIES FOR HGI PERFORMANCE 41
Recall the initial condition q r introduced in (2.2). Theorem 9.3.
Suppose ˆ q r . = q r / r satisfies sup r > ˆ q r < ∞ . Let { t r } be an increasing sequence suchthat t r ↑ ∞ as r → ∞ . Suppose that ˆ w r converges to some w ∈ R I + . Then, the random variables { θ rt r , r > are tight in the space P ¡ D ([0, 1] : R I + × R I ) ¢ .Proof. It suffices to show that the collection ©¡ ˆ W r ( t + · ), ˆ Z r ( t + · ) − ˆ Z r ( t ) ¢ , r > t > ª is tight in D ([0, 1] : R I + × R I ).Let F rt = σ ³ ˆ S rj ( ¯ B r ( s )), ˆ A rj ( s ) : j ∈ N J , 0 ≤ s ≤ t ´ , t ≥ j ∈ N both ˆ S rj ( ¯ B r ( t )) and ˆ A rj ( t ) are F rt -martingales. Consequently, there are κ , κ ∈ (0, ∞ ) such that for any r > δ > F rt -stopping times τ , τ satisfying τ ≤ τ ≤ τ + δ ≤ E h¡ ˆ Z ri ( τ ) − ˆ Z ri ( τ ) ¢ i ≤ κ J X j = G ri , j ³ E h ( ˆ A rj ( τ ) − ˆ A rj ( τ )) i + E h ( ˆ S rj ( ¯ B r ( τ )) − ˆ S rj ( ¯ B r ( τ ))) i´ ≤ κ J X j = E [ τ − τ ] + J X j = E £ ¯ B r ( τ ) − ¯ B r ( τ ) ¤ ≤ κ δ .This proves the tightness of the collection { ˆ Z r ( t + · ) − ˆ Z r ( t ), r > t > r ( K ̺ r − C ) → v ∗ , Theorem 9.2, and Lipschitz property of the Skorohodmap, to prove the tightness of { ˆ W r ( t + · ), r > t >
0} it now suffices to prove the tightness of{ ˆ W r ( t ), r > t > (cid:3) Recall that the reflected Brownian motion { ˇ W w } w ∈ R I + in (3.2) has a unique invariant proba-bility distribution which we denote as π . We will denote by Π the unique measure on C ([0, 1] : R I + )associated with this Markov process with initial distribution π . The following theorem gives a char-acterization of the weak limit points of the sequence θ rt r in Theorem 9.3. We denote the canonicalcoordinate processes on D ([0, 1] : R I + × R I ) as ( w ( t ), z ( t )) ≤ t ≤ . Let G t . = σ {( w ( s ), z ( s )) : 0 ≤ s ≤ t } bethe canonical filtration on this space. Theorem 9.4.
Suppose ˆ q r . = q r / r satisfies sup r > ˆ q r < ∞ . Also suppose that θ rt r converges in dis-tribution, along some subsequence as r → ∞ , to a P ( D ([0, 1] : R I + × R I )) valued random variable θ given on some probability space ( ¯ Ω , ¯ F , ¯ P ) . Then for ¯ P a.e. ω , under θ ( ω ) ≡ θ ω the following hold.(1) θ ω ( C ([0, 1] : R I + × R I )) = .(2) { z ( t )} ≤ t ≤ is a G t -Brownian motion with covariance matrix Σ = ΛΛ ′ , where Λ is as intro-duced above (3.2) .(3) {( w ( t ), z ( t ))} ≤ t ≤ satisfy θ ω a.s. w ( t ) = Γ ( w (0) − v ∗ ι + z )( t ), 0 ≤ t ≤ (4) θ ω ◦ ( w (0)) − = π and thus denoting the first marginal of θ ω on C ([0, 1] : R I + ) as θ ω , we have θ ω = Π . Proof.
For notational simplicity we denote the convergent subsequence of θ rt r by the same symbol.For ( x , y ) ∈ D ([0, 1] : R I + × R I ) define j ( x , y ) = sup ≤ t < °° ( x ( t ), y ( t )) − ( x ( t − ), y ( t − )) °° . Then there is a κ ∈ (0, ∞ ) such that for all r , E θ rt r (( x , y ) : j ( x , y ) > κ / r ) =
0. Thus in particular, for every δ ∈ (0, ∞ ),as r → ∞ , E θ rt r (( x , y ) : j ( x , y ) > δ ) →
0. By weak convergence of θ rt r to θ and Fatou’s lemma we thenhave E θ (( x , y ) : j ( x , y ) > δ ) = θ rt r (resp. θ ) as E θ rtr (resp. E θ ). Let f : D ([0, 1] : R I + × R I ) → R be a continuous and bounded function. We now argue that for all 0 ≤ s < t ≤
1, and i ∈ N I ¯ E ¡¯¯ E θ ¡ f ( w ( · ∧ s ), z ( · ∧ s ))( z i ( t ) − z i ( s )) ¢¯¯ ∧ ¢ =
0. (9.3)This will prove that { z ( t )} ≤ t ≤ is a G t -martingale under θ ω for a.e. ω . To see (9.3) note that E E θ rtr £ f ( w ( · ∧ s ), z ( · ∧ s ))( z i ( t ) − z i ( s )) ¤ = E · t r Z t r f ( ˆ W r ( u + ( · ∧ s )), ˆ Z r ( u + ( · ∧ s )) − ˆ Z r ( u ))[ ˆ Z ri ( u + t ) − ˆ Z ri ( u + s )] d u ¸ = t r Z t r Z u E ( H i ( u ) H i ( v )) d v d u ,where for u ≥ H i ( u ) = f ( ˆ W r ( u + ( · ∧ s )), ˆ Z r ( u + ( · ∧ s )) − ˆ Z r ( u ))( ˆ Z ri ( u + t ) − ˆ Z ri ( u + s )).Since ˆ Z ri is a martingale, we have for v < u − E ( H i ( u ) H i ( v )) =
0. Also from properties of Poissonprocesses it follows that for every p ≥ r > u ≥ s , t ∈ [0,1] E °° ˆ Z r ( u + t ) − ˆ Z r ( u + s ) °° p . = m p < ∞ . (9.4)Thus since f is bounded , we have for some κ ∈ (0, ∞ )2 t r Z t r Z u E ( H i ( u ) H i ( v )) d v d u ≤ κ t r → r → ∞ . Thus as r → ∞ ¯ E ³¯¯¯ E θ rtr ¡ f ( w ( · ∧ s ), z ( · ∧ s ))( z i ( t ) − z i ( s )) ¢¯¯¯ ∧ ´ → t ∈ [0, 1], sup r > E E θ rtr ( z i ( t )) < ∞ .In order to argue that { z ( t )} ≤ t ≤ is a G t -Brownian motion with covariance matrix Σ it now sufficesto show that defining m ( t ) . = z ( t ) z ′ ( t ) − t Σ , { m ( t )} ≤ t ≤ is a I dimensional { G t }-martingale. Oncemore, it suffices to show that with f as before, 0 ≤ s < t ≤
1, and i , l ∈ N I ,¯ E ¡¯¯ E θ ¡ f ( w ( · ∧ s ), z ( · ∧ s ))( m i , l ( t ) − m i , l ( s )) ¢¯¯ ∧ ¢ =
0. (9.5)For this note that¯
E E θ rtr £ f ( w ( · ∧ s ), z ( · ∧ s ))( m i , l ( t ) − m i , l ( s )) ¤ = E · t r Z t r f ( ˆ W r ( u + ( · ∧ s )), ˆ Z r ( u + ( · ∧ s )) − ˆ Z r ( u ))[ ˆ M r , ui , l ( t ) − ˆ M r , ui , l ( s )] d u ¸ = t r Z t r Z u E ( H ri , l ( u ) H ri , l ( v )) d v d u , (9.6)where for u ≥ M r , ui , l ( t ) = ¡ ˆ Z ri ( u + t ) − ˆ Z ri ( u ) ¢ ¡ ˆ Z rl ( u + t ) − ˆ Z rl ( u ) ¢ − t Σ il and H ri , l ( u ) = f ( ˆ W r ( u + ( · ∧ s )), ˆ Z r ( u + ( · ∧ s )) − ˆ Z r ( u ))[ ˆ M r , ui , l ( t ) − ˆ M r , ui , l ( s )]. ONTROL POLICIES FOR HGI PERFORMANCE 43
Write ˆ M r , ui , l ( t ) − ˆ M r , ui , l ( s ) = ˆ Ψ ri , l ( u ) + ¯ ξ ri , l ( u ),where ˆ Ψ ri , l ( u ) = ( ˆ Z ri ( u + t ) − ˆ Z ri ( u + s ))( ˆ Z rl ( u + t ) − ˆ Z rl ( u + s )) − J X j = G ri , j K l , j ( ¯ B rj ( u + t ) − ¯ B rj ( u + s ) + ( t − s ) ̺ rj )and ¯ ξ ri , l ( u ) = J X j = G ri , j K l , j ( ¯ B rj ( u + t ) − ¯ B rj ( u + s ) + ( t − s ) ̺ rj ) − ( t − s ) Σ i , l . (9.7)Then for 0 ≤ v ≤ u ≤ t r | E ( H ri , l ( u ) H ri , l ( v )) | ≤ | E ( ˆ H ri , l ( u ) ˆ H ri , l ( v )) | + k f k ∞ sup u ≥ E ( ¯ ξ ri , l ( u )) + k f k ∞ · sup u ≥ E ( ˆ Ψ ri , l ( u )) ¸ · sup u ≥ E ( ¯ ξ ri , l ( u )) ¸ , (9.8)where ˆ H ri , l ( u ) = f ( ˆ W r ( u + ( · ∧ s )), ˆ Z r ( u + ( · ∧ s )) − ˆ Z r ( u )) ˆ Ψ ri , l ( u ).From (9.4), for some κ ∈ (0, ∞ ) sup r > u , v > E | ˆ H ri , l ( u ) ˆ H ri , l ( v ) | ≤ κ .Also, from martingale properties of ˆ A j and ˆ S j we see that for v < u − E ( ˆ H ri , l ( u ) ˆ H ri , l ( v )) = r → ∞ t r Z t r Z u | E ( ˆ H ri , l ( u ) ˆ H ri , l ( v )) | d v d u ≤ κ t r →
0. (9.9)From (9.4) once more, we have for some κ ∈ (0, ∞ )sup u ≥ r > E ( ˆ Ψ ri , l ( u )) ≤ κ . (9.10)We now argue that sup u ≥ E ( ¯ ξ ri , l ( u )) → r → ∞ . (9.11)Note that once (9.11) is proved, it follows on combining (9.6), (9.8), (9.9) and (9.11) that E E θ rtr £ f ( w ( · ∧ s ), z ( · ∧ s ))( m i , l ( t ) − m i , l ( s )) ¤ → r → ∞ . Once more using the moment bound in (9.4) we then have (9.5) completing the proof of(2). We now return to the proof of (9.11). We note that for some κ ∈ (0, ∞ )sup u , r > | ¯ ξ ri , l ( u ) | ≤ κ a.s. .Thus for any ǫ ∈ (0, ∞ ) sup u > E | ¯ ξ ri , l ( u ) | ≤ ǫ + κ sup u > P ( | ¯ ξ ri , l ( u ) | > ǫ ). (9.12)Next from properties of Poisson processes it follows that for any ˜ ǫ ∈ (0, ∞ ), as r → ∞ sup u ≥ P ³¯¯¯ ¯ A rj ( u + t ) − ¯ A rj ( u + s ) − ( t − s ) λ rj ¯¯¯ > ˜ ǫ ´ → and sup u ≥ P ³¯¯¯ ¯ S rj ( ¯ B rj ( u + t )) − ¯ S rj ( ¯ B rj ( u + t )) − ( ¯ B rj ( u + t ) − ¯ B rj ( u + s )) µ rj ¯¯¯ > ˜ ǫ ´ → r → ∞ sup u ≥ P ³¯¯¯ ¯ A rj ( u + t ) − ¯ A rj ( u + s ) − ³ ¯ S rj ( ¯ B rj ( u + t )) − ¯ S rj ( ¯ B rj ( u + t )) ´¯¯¯ > ˜ ǫ ´ = sup u ≥ P ³¯¯¯ ¯ Q rj ( u + t ) − ¯ Q rj ( u + s ) ¯¯¯ > ˜ ǫ ´ → r → ∞ sup u ≥ P ³¯¯¯ ( t − s ) λ rj − ( ¯ B rj ( u + t ) − ¯ B rj ( u + s )) µ rj ¯¯¯ > ˜ ǫ ´ →
0. (9.13)Recalling the definition of ¯ ξ ri , l ( u ) from (9.7) and noting that 2 P Jj = G i j K l , j ̺ j = Σ il , we see from(9.13) that for any ǫ ∈ (0, ∞ ) sup u > P ( | ¯ ξ ri , l ( u ) | > ǫ ) → r → ∞ . Using this in (9.12) and sending ǫ → r ( K ̺ r − C ) → v ∗ as r → ∞ , we have for every t ∈ [0, 1], as r → ∞ E E θ rtr £°° w ( t ) − Γ ( w (0) + z − v ∗ ι )( t ) °° ∧ ¤ = t r Z t r E £°° ˆ W r ( u + t ) − Γ ( ˆ W r ( u ) + ˆ Z r ( u + · ) − ˆ Z r ( u ) − v ∗ ι )( t ) °° ∧ ¤ d u → θ rt r → θ in distribution, we have from continuous mapping theorem E E θ £°° w ( t ) − Γ ( w (0) + z + v ∗ ι )( t ) °° ∧ ¤ = g : R I + → R and t ∈ [0, 1] E ¯¯ E θ ( g ( w ( t ))) − E θ ( g ( w (0))) ¯¯ = r → ∞ E ¯¯¯ E θ rtr ( g ( w ( t ))) − E θ ( g ( w (0))) ¯¯¯ = E ¯¯¯¯ t r Z t r g ( ˆ W r ( u + t )) d u − t r Z t r g ( ˆ W r ( u )) d u ¯¯¯¯ ≤ k g k ∞ t r → θ rt r → θ and applying continuousmapping theorem. This completes the proof of the theorem. (cid:3)
10. P
ROOFS OF T HEOREMS
AND C . Proof of Theorem 4.7.
From Theorem 8.2 and noting that h · ˆ Q r ( t ) ≥ C ( ˜ W r ( t )) a.s., we have E t r Z t r | h · ˆ Q r ( t ) − C ( ˜ W r ( t )) | d t ≤ Br α − (1 + | ˆ q r | ). ONTROL POLICIES FOR HGI PERFORMANCE 45
Next, from Theorem 6.3 we see that C is a Lipschitz function. Let L C denote the correspondingLipschitz constant. Since M r → M , we can find η r ∈ (0, ∞ ) such that η r → r → ∞ and | ˜ W r ( t ) − ˆ W r ( t ) | ≤ η r | ˆ Q r ( t ) | for all t ≥ r >
0. (10.1)From Theorem 8.5 it then follows that, as r → ∞ E t r Z t r | C ( ˜ W r ( t )) − C ( ˆ W r ( t )) | d t ≤ L C η r t r Z t r E | ˆ Q r ( t ) | d t → t r Z t r C ( ˆ W r ( t )) → Z C ( w ) π ( d w ), in L , as r → ∞ . (10.2)From Theorems 9.3 and 9.4, for every L ∈ (0, ∞ ),1 t r Z t r C L ( ˆ W r ( t )) → Z C L ( w ) π ( d w ), in L , as r → ∞ where C L ( w ) . = C ( w ) ∧ L for w ∈ R I + . Also, from linear growth of C and Theorem 8.5, as L → ∞ ,sup r > t r Z t r E | C ( ˆ W r ( t )) − C L ( ˆ W r ( t )) | d t ≤ L sup r > t r Z t r E C ( ˆ W r ( t )) d t → R C ( w ) π ( d w ) < ∞ . Combining this with the abovetwo displays we now have (10.2) and the result follows. (cid:3) We now prove the convergence of the discounted cost. Proof is a simpler version of the argu-ment in the proof of Theorem 4.7 and therefore we omit some details.
Proof of Theorem 4.8.
Minor modifications of the proof of Theorem 9.3 together with Theorem9.2 show that for any T < ∞ ˆ W r converges in D ([0, T ] : R I + ) to ˇ W w . Thus using continuity of C , forevery L ∈ (0, ∞ ) and C L as in the proof of Theorem 4.7, for every T < ∞ ,lim r →∞ E ·Z T e − θ t C L ¡ ˆ W r ( t ) ¢ d t ¸ = E ·Z T e − θ t C L ¡ ˇ W w ( t ) ¢ d t ¸ .From Theorem 8.5 we have, as L → ∞ ,sup r > E Z ∞ e − θ t | C ( ˆ W r ( t )) − C L ( ˆ W r ( t )) | d t ≤ L sup r > Z ∞ e − θ t E C ( ˆ W r ( t )) d t → T → ∞ sup r > Z ∞ T e − θ t E C ( ˆ W r ( t )) d t → Z ∞ T e − θ t E C ( ˇ W w ( t )) d t → E R ∞ e − θ t C ( ˇ W w ( t )) d t < ∞ it then follows that for every T ∈ (0, ∞ ) E Z ∞ e − θ t h · ˆ Q r ( t ) d t → E Z ∞ e − θ t C ( ˇ W w ( t )) d t .The result follows. (cid:3) A PPENDIX
A. L
ARGE D EVIATION E STIMATES FOR P OISSON P ROCESSES
The following result gives classical exponential tail bounds for Poisson processes. For the proofof the first estimate we refer the reader to [12] while the second result is a consequence of [13,Section 4.1l1 3, Theorem 5].
Theorem A.1.
Let N r ( t ) be a Poisson process with rates λ r such that lim r →∞ λ r = λ ∈ (0, ∞ ) . Thenfor any ǫ ∈ (0, ∞ ) there exist B , B , R ∈ (0, ∞ ) such that for all < σ < ∞ and r ≥ R we haveP µ sup ≤ t ≤ ¯¯¯¯ N r ( σ t ) σ − λ r t ¯¯¯¯ > ǫ ¶ ≤ B e − σ B and for all T ∈ (0, ∞ ) P µ sup ≤ t ≤ T ¯¯ N r ( r t ) − r t λ r ¯¯ ≥ ǫ r T ¶ ≤ B e − B T A CKNOWLEDGEMENT
This research has been partially supported by the National Science Foundation (DMS-1305120),the Army Research Office (W911NF-14-1-0331) and DARPA (W911NF-15-2-0122).R
EFERENCES [1] B. Ata and S. Kumar,
Heavy traffic analysis of open processing networks with complete resource pooling: asymptoticoptimality of discrete review policies , The Annals of Applied Probability (2005), no. 1A, 331–391.[2] S. L Bell and R. J Williams, Dynamic scheduling of a system with two parallel servers in heavy traffic with resourcepooling: asymptotic optimality of a threshold policy , The Annals of Applied Probability (2001), no. 3, 608–649.[3] S. L Bell and R. J Williams, Dynamic scheduling of a parallel server system in heavy traffic with complete resourcepooling: Asymptotic optimality of a threshold policy , Electronic Journal of Probability (2005), 1044–1115.[4] V. Böhm, On the continuity of the optimal policy set for linear programs , SIAM Journal on Applied Mathematics (1975), no. 2, 303–306.[5] A. Budhiraja and A. P. Ghosh, A large deviations approach to asymptotically optimal control of crisscross network inheavy traffic , The Annals of Applied Probability (2005), no. 3, 1887–1935.[6] A. Budhiraja and A. P. Ghosh, Diffusion approximations for controlled stochastic networks: An asymptotic bound forthe value function , The Annals of Applied Probability (2006), no. 4, 1962–2006.[7] J. M. Harrison and R. J. Williams, Brownian models of open queueing networks with homogeneous customer popu-lations , Stochastics (1987), no. 2, 77–115. MR912049[8] J M. Harrison, Brownian models of open processing networks: Canonical representation of workload , Annals of Ap-plied Probability (2000), 75–103.[9] J M. Harrison, C. Mandayam, D. Shah, and Y. Yang,
Resource sharing networks: Overview and an open problem ,Stochastic Systems (2014), no. 2, 524–555.[10] J M. Harrison and J. A Van Mieghem, Dynamic control of brownian networks: state space collapse and equivalentworkload formulations , The Annals of Applied Probability (1997), 747–771.[11] W. Kang, F. Kelly, N. Lee, and R. Williams,
State space collapse and diffusion approximation for a network operatingunder a fair bandwidth sharing policy , The Annals of Applied Probability (2009), 1719–1780.[12] T. G. Kurtz,
Strong approximation theorems for density dependent Markov chains , Stochastic Process. Appl. (1978),223–240.[13] R. Sh. Liptser and A. N. Shiryayev, Theory of martingales , Mathematics and its Applications (Soviet Series), vol. 49,Kluwer Academic Publishers Group, Dordrecht, 1989.[14] L. Massoulie and J. W Roberts,
Bandwidth sharing and admission control for elastic traffic , Telecommunicationsystems15