Control of parallel non-observable queues: asymptotic equivalence and optimality of periodic policies
CControl of parallel non-observable queues: asymptotic equivalenceand optimality of periodic policies ∗ Jonatha Anselmi † , Bruno Gaujal ‡ and Tommaso Nesti § Abstract
We consider a queueing system composed of a dispatcher that routes jobs to a set of non-observablequeues working in parallel. In this setting, the fundamental problem is which policy should the dispatcherimplement to minimize the stationary mean waiting time of the incoming jobs. We present a structuralproperty that holds in the classic scaling of the system where the network demand (arrival rate of jobs)grows proportionally with the number of queues. Assuming that each queue of type r is replicated k times, we consider a set of policies that are periodic with period k (cid:80) r p r and such that exactly p r jobsare sent in a period to each queue of type r . When k → ∞ , our main result shows that all the policies inthis set are equivalent , in the sense that they yield the same mean stationary waiting time, and optimal ,in the sense that no other policy having the same aggregate arrival rate to all queues of a given type cando better in minimizing the stationary mean waiting time. This property holds in a strong probabilisticsense. Furthermore, the limiting mean waiting time achieved by our policies is a convex function of thearrival rate in each queue, which facilitates the development of a further optimization aimed at solvingthe fundamental problem above for large systems. In computer and communication networks, the access of jobs to resources (web servers, network links, etc.)is usually regulated by a dispatcher. A fundamental problem is which algorithm should the dispatcherimplement to minimize the mean delay experienced by jobs. There is a vast literature on this subject andthe structure of the optimal algorithm strongly depends on i) the information available to the dispatcher, ii) the topology of the network and iii) how jobs are processed by resources. We are interested in a scenariowhere: • The dispatcher has static information of the system; • The network topology is parallel; • Resources process jobs according to the first-come-first-served discipline.Static information means that the dispatcher knows the probability distributions of job sizes and inter-arrivaltimes but cannot observe the dynamic state of resources such as the current number of jobs in their queues.This scenario can be of interest in the context of volunteer computing, cloud computing, web server farms,etc.; see, e.g., [24, 25, 20] respectively.In this framework, the problems of finding an algorithm, or policy , that minimizes the mean stationarydelay and of determining the minimum mean stationary delay are both considered difficult; see, e.g., [2, 1] foran overview. A policy can be defined as a function that maps a natural number n , corresponding to the n -th ∗ Research partially supported by grant SA-2012/00331 of the Department of Industry, Innovation, Trade and Tourism(Basque Government) and grant MTM2010-17405 (Ministerio de Ciencia e Innovaci´on, Spain) which sponsored an internshipof T. Nesti at BCAM. † Basque Center for Applied Mathematics (BCAM), Al. de Mazarredo 14, 48009 Bilbao, Basque Country, Spain; INRIABordeaux Sud Ouest, 200 av. de la Vieille Tour, 33405 Talence, France. Email: [email protected] ‡ INRIA and LIG, Zirst 51, Av. J. Kuntzmann, MontBonnot Saint-Martin, 38330, France. Email: [email protected] § Basque Center for Applied Mathematics (BCAM), Al. de Mazarredo 14, 48009 Bilbao, Basque Country, Spain; Dipartimentodi Matematica, University of Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy. Email: [email protected] a r X i v : . [ c s . PF ] J a n ob arriving to the dispatcher, to a probability mass function P n over the set of resources. When the n -th jobarrives, the dispatcher sends it to resource i with probability P n ( i ). Unfortunately, the problem of findingan optimal policy is intractable and for this reason two extreme families of policies have received particularattention in the literature: probabilistic policies, obtained when P n is constant (in n ), and deterministic policies, obtained when P n puts the whole mass on a single resource.When dealing with probabilistic policies, the difficulty of the problem is simplified by the fact that thearrival process at each resource is a renewal process, provided that the same holds for the arrival process atthe dispatcher. This allows one to decompose the problem and, using the theory of the mean waiting time ofthe single GI/GI/1 queue, to immediately reduce it to a relatively simple optimization problem. This problemis usually convex and there exist efficient numerical procedures for their solution; e.g., [30, 34, 32, 13, 14, 8].Contrariwise, when dealing with deterministic policies, one of the main difficulties is that the arrivalprocess at each resource is hardly ever a renewal process. This prevents one from decomposing the problemand directly using the classic theory of the single queue as it has been done for probabilistic policies. Giventhis difficulty, researchers divided this problem in two subproblems: i) In the first subproblem, the optimal deterministic policy is searched among all the deterministic policiesensuring that the long-term fractions of jobs to be sent to each resource is kept fixed (denote suchfractions by vector p ); ii) In the second subproblem, the output of the first subproblem is employed to develop a further opti-mization over p .In this paper, we focus on the first subproblem and, under some system scaling, we identify a set of policiesthat are optimal. This result is used to reduce the second subproblem to the solution of a convex optimization.One of the folk theorem of queueing theory says that determinism in the inter-arrival times minimizesthe waiting time of the single queue [18, 23]. In view of this classic insight and fixing fractions p , it isnot surprising that an optimal policy tries to make the arrival process at each resource as regular (or lessvariable) as possible. Thus, our stochastic scheduling problem can be essentially converted into a problemin word combinatorics. If the dispatcher must ensure fractions p , the main result known in the literature isthat balanced sequences are optimal admission sequences [19, 1]. However, balanced sequences of given rates p are known to exist in very few particular cases. These cases are captured by Fraenkel’s conjecture, whichis still open to the best of our knowledge [36]; see also [2, Chapter 2], which contains an overview of whichrates p are balanceable.Matter of fact, the problem of finding an optimal deterministic policy is still considered difficult [10, 3,38, 22, 2, 7, 21]. The only exceptions are when resources are stochastically equivalent, where round-robin is known to be optimal in a strong sense [27], or when the dispatcher routes jobs to two resources, wherebalanced sequences can be always constructed no matter the value of rates p [19]. In presence of more thantwo queues, we stress that balanced sequences with given rates p do not exist in general. This non-existencemakes the problem difficult and one still wonders which structure should an optimal policy have when p is not balanceable. When the routing is performed to two resources, jobs join the dispatcher following aPoisson process and service times have an exponential distribution, the optimal rates p as function of theinter-arrival and service times have a fractal structure, see [16, Figure 8]. This puts further light on thecomplexity of the problem even in a simple scenario.While deterministic policies are believed to be more difficult to study than probabilistic policies (determin-istic and probabilistic in the sense described above), they can achieve a significantly better performance [3].This holds also for the variance of the waiting time because, as discussed above, the arrival process at eachresource is much more regular in the deterministic case, especially if there are several resources as we showin this paper. A particular class of deterministic policies, namely billiard policies, have been recently imple-mented in the context of large volunteer and cloud computing to improve the performance of real applicationssuch as SETI@home [24, 25]. In the framework described above, we are interested in deriving structural properties of deterministic policieswhen the system size is large. We study a scaling of the system where the arrival rate of jobs, λk , grows to Round-robin sends the n -th job to resource ( n mod R ) + 1, where R is the total number of resources. Rk , while keeping the networkload (or utilization) fixed. This scaling is often used in the queueing community. Specifically, there are R types of queues, and k is the number of queues in each type, i.e., the parameter that we will let grow toinfinity. Beyond issues related to the tractability of the problem, this type of scaling is motivated by the factthat the size of real systems is large and that replication of resources is commonly used to increase systemreliability.First, with respect to a class of periodic policies, we define the random variable of the waiting time ofeach incoming job. This is done using Lindley’s equation [26] and a suitable initial randomization. Usingsuch randomization, we can adapt the framework developed by Loynes in [28] to our setting where jobs aresent to a set of parallel queues. In particular, Theorem 1 shows the monotone convergence in distributionof the waiting time of each incoming job. Then, with respect to a given vector p ∈ N R , we define a certainsubset of policies that are periodic with period k (cid:80) r p r and such that exactly p r jobs are sent in a period toeach queue of type r . While further details will be developed in Section 3.1, this set is meant to imply thatqueues of a given type are visited in a round-robin manner and that arrivals are “well distributed” amongthe different queue types. When k → ∞ , our main result states that all the policies in this set are equivalent ,in the sense that they yield the same mean stationary waiting time, and optimal , in the sense that no otherpolicy having the same aggregate arrival rate to all queues of a given type can do better in minimizing themean stationary waiting time. In particular, we show that the stationary waiting time converges both indistribution and in expectation to the stationary waiting time of a system of independent D/GI/1 queueswhose parameters only depend on p , λ and the distribution of the service times. This is shown in Theorems 2and 3, respectively.The main idea underlying our proof stands in analyzing the sequence of stationary waiting times alongappropriate subsequences. Along these subsequences, it is possible to extract a pattern for the arrival processof each queue that is common to all members of the subsequence. Such pattern is exploited to establishmonotonicity properties in the language of stochastic orderings. These properties hold for the consideredsubsequences only: they do not hold true along any arbitrary subsequence and counterexamples can begiven. These properties will imply the uniform integrability of the sequence of stationary waiting times andwill allow us to work on expected values.Summarizing, fixing the proportions p of jobs to send to each queue type and given k large, our resultsstate that all the policies belonging to the set that we identify in Section 3.1 yield the same asymptoticperformance and are asymptotically optimal. Furthermore, using known properties of the D/GI/1 queue, weobtain that the stationary mean waiting time obtained in our limit is a convex function of p . This reduces thecomplexity of subproblem ii) above because it boils down to the solution of a convex optimization problem.This paper is organized as follows. Section 2 introduces the model under investigation and provides acharacterization of the stationary waiting time (Theorem 1); Section 3 introduces a class of policies andpresents our main results (Theorems 2 and 3); Section 4 is devoted to proofs; finally, Section 5 draws theconclusions of this paper. We consider a queueing system composed of R types of queues (or resources, servers) working in parallel.Each queue of type r is replicated k times, for all r = 1 , . . . , R , so there are kR queues in total. Parameter k is a scaling factor and we will let it grow to infinity. The service discipline of each queue is first-come-first-served (FCFS) and the buffer size of each queue is infinite. A stream of jobs (or customers) joins thequeues through a dispatcher. The dispatcher routes each incoming job to a queue according to some policyand instantaneously. Figure 1 illustrates the structure of the queueing model under investigation. In thefollowing, indices r , κ , n will be implicitely assumed to range from 1 to R , from 1 to k , in N , respectively.All the random variables that follow will be considered belonging to a fixed underlying probabilitytriple (Ω , F , Pr).Let ( T ( k ) n ) n ∈ N and ( S ( k ) n,κ,r ) n ∈ N be given sequences of i.i.d. random variables in R +2 . These sequences areall assumed to be independent each other. Quantity T ( k ) n is interpreted as the inter-arrival time between the For any E ⊆ R , we let E + def = { x ∈ E : x > } . Type 1 μ Type 1 μ Type 1 μ R ... Type R μ R Type R μ R Type R
Dispatcher λ k jobs/time k queue sk queue s π Figure 1: Structure of the parallel queueing model under investigation. n -th and the ( n + 1)-th jobs arriving to the dispatcher. Quantity S ( k ) n,κ,r is interpreted as the service times ofthe n -th job arriving at the κ -th queue of type r . We assume that S ( k )1 ,κ,r = st S ( k )1 , ,r and that E S ( k ) n,κ,r = µ − r .For the arrival process at the dispatcher, we will refer to the following cases. Case 1.
The process ( T ( k ) n ) n ∈ N is a renewal process with rate λk and such that Var T ( k ) n = o (1 /k ) . Case 2.
The process ( T ( k ) n ) n ∈ N is a Poisson process with rate λk . Case 3.
The process ( T ( k ) n ) n ∈ N is constant with rate λk , i.e., T ( k ) n = ( λk ) − . It is clear that Cases 2 and 3 are both more restrictive than Case 1.Let (cid:107) · (cid:107) denote the L -norm.Let q def = ( q r,κ ) ∈ Q R + × Q k + be such that (cid:107) q (cid:107) = 1. Quantity q r,κ will be interpreted as the proportion ofjobs sent to queue ( r, κ ).Let n ∗ def = n ∗ ( q ) def = min { n ∈ Z + : n q ∈ Z R + × Z k + } . Since q is a vector of rational numbers, n ∗ < ∞ .Let V be a discrete random variable with values in { , . . . , n ∗ } such that Pr( V = i ) = 1 /n ∗ , for all i = 1 , . . . , n ∗ . We assume that V is independent of any other random variable.Let A q ( k ) be the set of all functions π : N → { , . . . , R } × { , . . . , k } such that for all r and κq r,κ = 1 n ∗ n ∗ + V − (cid:88) n = V { π ( n )=( r,κ ) } and π ( n ) = π ( n + n ∗ ) (1)for all n , where E denotes the indicator function of event E . Thus, these functions are periodic with period n ∗ and n ∗ q r,κ is the number of jobs sent in a period to queue ( r, κ ). We refer to each element π ∈ A q ( k )as a policy (or a q -policy) operated by the dispatcher, and it is interpreted as follows: π ( n ) = ( r, κ ) meansthat the n -th job arriving to the dispatcher is sent to the k -th queue of type r if n ≥ V , otherwise it meansthat the n -th job is discarded. Thus, the outcome of random variable V gives the index of the first job thatis actually served by some queue. In other words, π ( V ) is the first queue that serves some job.Let ( T ( k ) n,κ,r ( π )) n ∈ N be the sequence of inter-arrival times that are induced by policy π at the κ -th queueof type r (under any of the cases above). By construction, T ( k ) n,κ,r ( π ) is the sum of a deterministic number ofinter-arrival times seen at the dispatcher. The arrival process ( T ( k ) n,κ,r ( π )) n ∈ N can be made stationary if it isallowed a shift in time and a suitable randomization (independent of V and of any other random variable) forthe inter-arrival time of the first arrival of each queue. We assume for now that this has been done (detailswill be given at the beginning of Section 4). As done in [28] and according to [15, p. 456], this implies that4e can extend the stationary process ( T ( k ) n,κ,r ( π )) n ∈ N to form a stationary process ( T ( k ) n,κ,r ( π )) n ∈ Z (clearly, thesame holds for the process ( S ( k ) n,κ,r ) n ∈ N ).The waiting time of the n -th job arriving to the κ -th queue of type r induced by a policy π ∈ A ( k ) isdenoted by W ( k ) n,r,κ ( π ). It is the time between its arrival at the dispatcher (or equivalently at the queue) andthe start of its service, and it is defined as follows: for n = 0, W ( k ) n,r,κ ( π ) = 0 and for n > W ( k ) n,r,κ ( π ) def = (cid:16) W ( k ) n − ,r,κ ( π ) + S ( k ) n,κ,r − T ( k ) n,κ,r ( π ) (cid:17) + , (2)where x + def = max { x, } . Equation (2) is known as Lindley’s recursion [26]. The assumption that W ( k )0 ,r,κ ( π ) =0 serves to avoid technicalities . It is known that the sequence of random variables ( W ( k ) n,r,κ ( π )) n ∈ N convergesin distribution to the random variable W ( k ) r,κ ( π ) def = (cid:32) sup n ≥ n (cid:88) n (cid:48) =0 S ( k ) − n (cid:48) ,κ,r − T ( k ) − n (cid:48) ,κ,r ( π ) (cid:33) + (3)and that W ( k ) n,r,κ ( π ) ≤ st W ( k ) n +1 ,r,κ ( π ), where ≤ st denote the usual stochastic order; see [28]. We refer to W ( k ) r,κ ( π ) as the stationary waiting time of jobs at the κ -th queue of type r .Given π , let f : N → N ×{ , . . . , R }×{ , . . . , k } be a mapping with the following meaning: f ( n ) = ( n (cid:48) , r, κ )means that the n -th job arriving to the dispatcher is the n (cid:48) -th customer joining queue ( r, κ ). If n < V , nojob is sent to any queue and thus we assume f ( n ) = 0. Note that f ( n ) is a deterministic function of randomvariable V . Since V is uniform over { , . . . , n ∗ } , for all n ≥ n ∗ we havePr( f ( n ) = r, f ( n ) = κ ) = q r,κ , (4)where f j refers to the j -th component of f . With this notation, quantity W ( k ) f ( n ) ( π ) is the waiting time of the n -th job arriving to the dispatcher induced by a policy π ∈ A q ( k ), for all n ≥ n ∗ .Now, let ( Q r,κ ) ∀ r,κ denote a partition of set { , . . . , n ∗ } . The subsets Q r,κ , for all r and k , are thusdisjoint and we further require that the number of points in Q r,κ is n ∗ q r,κ . Let W ( k ) ( π ) def = R (cid:88) r =1 k (cid:88) κ =1 { V ∈ Q r,κ } W ( k ) r,κ ( π ) . (5)Since V is independent of the W ( k ) r,κ ( π )’s, the distribution of W ( k ) ( π ) is the finite mixture of the W ( k ) r,κ ( π )’swith weights q . Next theorem says that W ( k ) ( π ) can be interpreted as the right random variable describingthe stationary waiting time of jobs achieved with policy π ∈ A q ( k ). It is proven by adapting the frameworkdeveloped by Loynes in [28]. In the remainder of the paper, convergence in distribution (in probability) isdenoted by d −→ (respectively, Pr −→ ). Theorem 1.
Let Case 1 hold. Let q be such that q r,κ λ < µ r for all r, κ and π ∈ A q ( k ) . Then, W ( k ) f ( n ) ( π ) ≤ st W ( k ) f ( n +1) ( π ) (6) and W ( k ) f ( n ) ( π ) d −−−−→ n →∞ W ( k ) ( π ) . (7)By using the monotone convergence theorem, Theorem 1 implies that also the moments of W ( k ) f ( n ) ( π )converge to the moments of W ( k ) ( π ), provided that they are finite.Finally, for a given p ∈ R R + , we define the auxiliary random variable W r ( p ), which corresponds to thestationary waiting time of a D/GI/1 queue with inter-arrival times ( T n,r ( p )) n ∈ N where T n,r ( p ) = (cid:107) p (cid:107) / ( p r λ ) Using a standard coupling argument and that each queue will empty in finite time almost surely, what follows can begeneralized easily to the case where W ( k )0 ,r,κ ( π ) ≥ S n, ,r ) n ∈ N , for all r . Let also V def = V ( p ) be a random variable with values in { , . . . , R } independent of any other random variable and such that Pr( V = r ) = p r / (cid:107) p (cid:107) , for all r , and let W ( p ) def = R (cid:88) r =1 { V = r } W r ( p ) . (8)Note that E W ( p ) = (cid:80) r p r (cid:107) p (cid:107) E W r ( p ) can be interpreted as the mean waiting time of R independent D/GI/1queues averaged over weights p/ (cid:107) p (cid:107) .We will be interested in establishing convergence results for W ( k ) when k → ∞ . With respect to somepolicies to be defined, we will show forms of convergence to W ( p ). Some remarks about the model above and Theorem 1 follow: • To agree on a definition for E W ( k ) or other moments of W ( k ) , one does not necessarily need to knowthe distribution of W ( k ) , e.g., one can achieve that using Cesaro sums [2]. Matter of fact, existingworks agree on the structure of E W ( k ) without constructing the distribution of W ( k ) . On the otherhand, our approach needs to know the distribution of W ( k ) (and thus Theorem 1) because we canprove convergence results for E W ( k ) only through a distributional convergence argument, that is [11,Theorem 3.5, pp. 31]. In particular, to prove E W ( k ) converges to E W ( p ), we will prove convergence indistribution of W ( k ) to W ( p ) and then the uniform integrability of the sequence of the W ( k ) ’s. This isthe reason why we need the characterization of the distribution of the stationary waiting time W ( k ) ,which we give in Theorem 1 and prove through classical arguments. • Though several works focused on finding policies that minimize the expected value E W ( k ) (see theintroduction), the analysis of E W ( k ) in our scaling where k → ∞ seems new. • We assume that q is a vector of rational numbers. From a practical standpoint, this is not a loss ofgenerality for obvious reasons. As an additional remark to support this assumption, we note that it hasbeen proven that in several cases the q -vector that minimize min π ∈A q ( k ) E W ( k ) ( π ) is indeed rational[2, Theorem 32, p.136]; see also [37].Our approach needs this assumption to prove (7). In the case where π ( n ) is not periodic, a casethat we do not consider here, and the limit lim n →∞ n (cid:80) nn (cid:48) =1 { π ( n (cid:48) )=( r,κ ) } does not exist, it would beinteresting to know whether some convergence in distribution of W ( k ) f ( n ) ( π ) occurs. A totally differentargument will be needed here. • The fact that in Case 1 we require that Var T ( k ) n = o ( k ) is not a loss of generality for Theorem 1, aswe do not let k → ∞ there. Case 1 covers the case where T ( k ) n has a ( Gk, α )-phase-type distributionwhere both G and α are not functions of k . • We may have assumed that each queue of type r was replicated kz r + o ( k ) times instead of just k times.This is essentially equivalent to our setting and our proofs can be easily adapted to this case thoughwe do not do it for simplicity. For p ∈ Z R + , let P p ( k ) def = (cid:40) q ∈ Q R + × Q k + : k (cid:88) κ =1 q r,κ k = p r (cid:107) p (cid:107) , ∀ r (cid:41) (9)and A p ( k ) def = (cid:91) q ∈P p ( k ) A q ( k ) . (10)6he set A p ( k ) is interpreted as the set of all periodic policies for which λk p r (cid:107) p (cid:107) is the aggregate mean arrivalrate of jobs to all type- r queues. We consider the following problem. Problem 1.
Let p ∈ Z R + be given. Determine the optimizers and the optimal objective function value of min π ∈A p ( k ) E W ( k ) ( π ) . (11)As discussed in the introduction, this is considered as a difficult problem. We are interested in establishingstructural properties of Problem 1 when k is large. In the following, we define a certain class of policies andthen present our main results. For p ∈ Z R + , we define C ( k ) p as the subset of all policies π ∈ A p ( k ) that satisfy the following properties: • The sequence ( π ( n )) n ∈ N has period (cid:107) p (cid:107) and p r is the number of jobs sent to type- r queues per period,for 1 ≤ r ≤ R . • Let n , n , . . . be the subsequence of all jobs that are sent to queues of type r . Then, π ( n ) = 1, and π ( n j +1 ) = π ( n j ) + (0 ,
1) if π ( n j ) < k and π ( n j +1 ) = ( r,
1) otherwise.The first property specifies the periodicity of π with respect the types of queues, while the second withrespect to queues. Thus, the second property says that queues of type r are accessed by jobs in a round-robin(or cyclic) order starting from queue 1, and we need it to ensure that the cardinality of C ( k ) p , i.e., |C ( k ) p | , doesnot vary with k . One can verify that |C ( k ) p | = (cid:107) p (cid:107) ! (cid:81) r ( p r !) . (12)These policies can be implemented in a distributed manner. More precisely, one can think that there aretwo tiers of dispatchers: the dispatcher in the first tier schedules jobs inter -group, while the dispatchers inthe second tier, R in total, schedule jobs intra -group and implement round-robin.With respect to a sequence of policies ( π ( k ) ) k ∈ N , where π ( k ) ∈ C ( k ) p , we will show (Lemma 2) T ( k ) n,κ,r ( π ( k ) ) Pr −−−−→ k →∞ (cid:107) p (cid:107) p r λ , ∀ n. (13)This means that the finite dimensional distributions of the arrival process at each queue will be ‘close’ to thedeterministic process, when k is large, which implies that some form of convergence to W ( p ) should occurin view of the continuity of the stationary waiting time [12]. Next theorem proves a first form of convergence of W ( k ) ( π ( k ) ) to W ( p ). Theorem 2.
Let Case 1 hold. Let p ∈ Z R + be such that λ p r (cid:107) p (cid:107) < µ r for all r . Let also an arbitrary sequence ( π ( k ) ) k ∈ N be given where π ( k ) ∈ C ( k ) p for all k . Then W ( k ) ( π ( k ) ) d −−−−→ k →∞ W ( p ) . (14)Theorem 2 is not enough to claim that E W ( k ) ( π ( k ) ) converges as well. Convergence of the first momentis important from an operational standpoint, as in practice one desires to optimize over E W ( k ) or Var W ( k ) .Under some additional assumptions, next theorem states that also the expected value and the variance of W ( k ) converge. Furthermore, it states that all the policies in set C ( k ) p are both asymptotically equivalent andasymptotically optimal, with respect to the criterion in Problem 1.7 heorem 3. Let Case 2 or 3 hold. Let p ∈ Z R + be such that λ p r (cid:107) p (cid:107) < µ r for all r . Let also an arbitrarysequence ( π ( k ) ) k ∈ N be given where π ( k ) ∈ C ( k ) p for all k . If E [( S ( k )1 ,κ,r ) ] < ∞ , then lim inf k →∞ min π ∈A p ( k ) E W ( k ) ( π ) = lim k →∞ E W ( k ) ( π ( k ) ) (15)lim k →∞ E W ( k ) ( π ( k ) ) = E W ( p ) . (16) Furthermore, if E [( S ( k )1 ,κ,r ) ] < ∞ , then lim k →∞ Var W ( k ) ( π ( k ) ) = (cid:88) r p r (cid:107) p (cid:107) (cid:16) Var W r ( p ) + ( E W r ( p ) − E W ( p )) (cid:17) . (17)Provided that k is large, thus, no other policy in A p ( k ) \ C ( k ) p can do better than any π ( k ) ∈ C ( k ) p tominimize E W ( k ) . It also explicits the limiting value of E W ( k ) ( π ( k ) ), which is E W ( p ). It is known that E W ( p ) is a convex function in p (e.g., [30]).To prove Theorem 3, we use the well-known fact that [5] W r ( p ) ≤ icx W ( k ) r,κ ( π ) (18)for any π ∈ A p ( k ), where ≤ icx denotes the increasing-convex order (see, e.g., [33, 35] for their definition).Using this lower bound first and then that the waiting time of the D/GI/1 queue is convex increasing in itsarrival rate, it is not difficult to show that E W ( p ) ≤ E W ( k ) ( π ) . (19)Then, we prove that the sequence E W ( k ) ( π ( k ) ) is upper bounded by a sequence that converges to the lowerbound in (19). An observation here is that the lower bound (18) holds under conditions that are weaker thanthose assumed in this paper; see [23]. For instance, it is possible to extend (19) (and thus Theorem 3) tothe case where i) policies are not periodic, ii) the fractions of jobs to send in each queue are not necessarilyrational numbers, and iii) policies are randomized [31], that is the case where π ( n ) is any probability massfunction over the set of queues. We do not investigate these extensions in further detail. In this section, we develop proofs for Theorems 1, 2 and 3. Before doing this, we fix some additional notationand show how it is possible to make the arrival process at each queue stationary with respect to any policyin π ( k ) ∈ A q ( k ), q ∈ Q kR + such that (cid:107) q (cid:107) = 1 (as assumed in Section 2).Let us consider the κ -th queue of type r and its arrival process ( T ( k ) n,κ,r ) n ∈ N . Each inter-arrival time clearlydepends on the policy π ( k ) implemented by the dispatcher, i.e., T ( k ) n,κ,r = T ( k ) n,κ,r ( π ( k ) ), though in the followingwe drop such dependence for notational simplicity. Since π ( k ) is periodic by construction with period n ∗ and n ∗ q r,κ jobs have to be sent within a cycle to the queue identified by the couple ( r, κ ), the sequence( T ( k ) n,κ,r ) n ∈ N is composed of a repeated pattern of n ∗ q r,κ inter-arrival times, that we can write as A ( k )1 ,κ,r , A ( k )2 ,κ,r , . . . , A ( k ) n ∗ q r,κ ,κ,r , (20)where each quantity A ( k ) j,κ,r , j = 1 , . . . , n ∗ q r,κ , is the sum of a deterministic number (that depends on π ( k ) )of inter-arrival times to the dispatcher. We denote such number by a ( k ) j,r,κ . Thus, A ( k ) j,κ,r = st a ( k ) j,r,κ (cid:88) n =1 T ( k ) n , (21)8here = st denotes equality in distribution.If π ( k ) ∈ C ( k ) p for some p ∈ Z R + , then we notice that a ( k ) j,r,κ does not vary with κ because by symmetrythe arrival processes of all queues of a given type are equal, in distribution, up to a shift in time. In thiscase, the arrival process at any queue of type r becomes a sequence composed of a repeated pattern of p r inter-arrival times that we can write as A ( k )1 ,κ,r , A ( k )2 ,κ,r , . . . , A ( k ) p r ,κ,r . (22)Therefore, when π ( k ) ∈ C ( k ) p , we will just write a ( k ) j,r instead of a ( k ) j,r,κ and it is also clear that p r (cid:88) j =1 a ( k ) j,r = k (cid:107) p (cid:107) (23)and that E T ( k ) n,κ,r = p r p r (cid:88) j =1 a ( k ) j,r E T ( k ) n = p r p r (cid:88) j =1 a ( k ) j,r kλ = (cid:107) p (cid:107) p r λ . (24)Now, we want to make ( T ( k ) n,κ,r ) n ∈ N stationary. This can be done as follows by randomizing over thefirst inter-arrival time of queue ( r, κ ). Now, let us consider the auxiliary random variables U r,κ , for all r and κ , which we assume independent each other and of any other random variable and having a uniformdistribution in [0 , T ( k )1 ,r,κ def = n ∗ q r,κ (cid:88) j =1 A ( k ) j,r,κ (cid:110) U r,κ ∈ (cid:104) j − n ∗ qr,κ , jn ∗ qr,κ (cid:105)(cid:111) (25)Therefore, if T ( k )1 ,r,κ = A ( k ) j,r,κ , for some j < n ∗ q r,κ , then T ( k )2 ,r,κ = A ( k ) j +1 ,r,κ and so forth according to thepattern (20). Defined in this manner, one can see that ( T ( k ) n,r,κ ) n ∈ N is stationary as desired. At this point,the issue is the following. Consider two queues, say ( r, κ ) and ( r (cid:48) , κ (cid:48) ), and suppose that T ( k )1 ,r,κ = A ( k ) j,r,κ and T ( k )1 ,r (cid:48) ,κ (cid:48) = A ( k ) j (cid:48) ,r (cid:48) ,κ (cid:48) . We should check whether policy π ( k ) is actually able to induce arrival processes at thequeues equal (samplepath-wise) to the ones built above through (25). One can easily see that this can bedone with a possible shift of time for the arrival process at the queues and possibly discarding a finite numberof jobs. This is allowed because these operations do not change the stationary behavior.In the remainder, we will use stochastic orderings. We will denote by ≤ st , ≤ cx and ≤ icx , the usualstochastic order , the convex order and the increasing convex order , respectively; we point to, e.g., [33, 35]for their definition.We will also refer to the following lemma, which can be easily proven. Lemma 1.
Let N be a finite positive integer and suppose ( f k,n ) ( k,n ) ∈ N ×{ ,...,N } is a semi-infinite array ofnumbers such that for some constant c , lim k →∞ f k,n = c , for n ∈ { , . . . , N } . Then, for any sequence ( n k ) k ∈ N with values in { , . . . , N } , lim k →∞ f k,n k = c . We now give proofs for our results, i.e., Theorems 1, 2 and 3.
Let random variables V and V be given such that V = st V and V = V − V > V = n ∗ . Weprove (6) through Strassen’s theorem building a coupling ( ˜ W ( k ) f ( n ) ( π ) , ˜ W ( k ) f ( n +1) ( π )) of W ( k ) f ( n ) ( π ) and W ( k ) f ( n +1) ( π )through V and V ensuring that ˜ W ( k ) f ( n ) ( π ) ≤ ˜ W ( k ) f ( n +1) ( π ). This is done as follows. First, let ˜ W ( k ) f ( n ) ( π ) and˜ W ( k ) f ( n +1) ( π ) be the Loynes waiting times (see [28, p. 501]) obtained when the first queue to serve a job isgiven by the outcome of V and V , respectively; thus, ˜ W ( k ) f ( n ) ( π ) is the waiting time at time 0 with n (cid:48) jobs inthe past at queue ( r, κ ), provided that f ( n ) = ( n (cid:48) , r, κ ). Then, we let the (Loynes) waiting times be drivenby the same realizations of the random inter-arrival and service times.9e now prove (7). Since π is periodic with period n ∗ , we first observe that π returns the same queuealong subsequence ( n n ∗ + i ) n ∈ N , for all i = 1 , . . . , n ∗ . Similarly, also the second and third component of f ( n n ∗ + i ) do not change along these subsequences, though they are not known in advance because theydepend on the outcome of random variable V , see (4). Thus, for all i = n ∗ + 1 , . . . , n ∗ , by construction wehave Pr( f ( n n ∗ + i ) = r, f ( n n ∗ + i ) = κ ) = Pr( f ( i ) = r, f ( i ) = κ ) = q r,κ , (26)and we get lim n →∞ Pr( W ( k ) f ( nn ∗ + i ) ( π ) ≤ t ) (27a)= lim n →∞ (cid:88) r,κ q r,κ Pr( W ( k ) n,r,κ ( π ) ≤ t | ( f ( i ) , f ( i )) = ( r, κ )) (27b)= (cid:88) r,κ q r,κ Pr( W ( k ) r,κ ( π ) ≤ t ) (27c)= Pr( W ( k ) ( π ) ≤ t ) . (27d)In (27b), we have conditioned on f ( i ). In (27c), we have used that W ( k ) n,r,κ ( π ) converges in distribution to W ( k ) r,κ ( π ); see [28]. In (27d), we have used the definition of W ( k ) ( π ). Now, since the limit in (27d) does notdepend on i , the proof is concluded by applying Lemma 1 once noted that n ∗ is a finite positive integer. We first observe that we can prove this theorem under some assumption on the sequence ( π ( k ) ) k ∈ N . Given π (1) ∈ C (1) p , we require that for all k : π ( k )1 ( n ) = π (1)1 ( n ) , ∀ n. (28)One may refer to these sequences as the ‘natural’ scaling of policy π (1) : in the two-tier interpretation of ourpolicies, (28) means that the dispatcher at the first tier implements the same policy, to queue types, when k grows.These sequences will be assumed along this proof. If this theorem holds for these sequences, then it alsoholds for all the sequences in view of Lemma 1 and of the fact that the cardinality of C ( k ) p does not varywith k .For m ∈ N , let k m def = m lcm( p ) , (29)where lcm( p ) denotes the least common multiple of p , . . . , p R . The subsequences ( k m + i ) m ∈ N , for all i = 1 , . . . , lcm( p ), play a key role in our proof of Theorem 3. Along these subsequences, next fact holds trueand follows by construction of the policies in set C ( k ) p : it is a direct consequence of the fact that queues ofthe same type are visited in a round-robin manner. Fact 1.
For m > , j = 1 , . . . , p r and i ∈ N , a ( p r m + i ) j,r = a ( p r ( m − i ) j,r + (cid:107) p (cid:107) .Proof. By construction, we have π ( p r m + i )2 ( V ) = 1 and π ( p r m + i ) ( V ) = π ( p r ( m − i ) ( V ), which in some sensecouples the arrival processes at queues of the ( p r m + i )-th and ( p r ( m −
1) + i )-th systems.Without loss of generality, let us assume that ( r,
1) is the queue that receives the first job.Now, Fact 1 holds true because ( p r m + i ) − ( p r ( m −
1) + i ) = p r jobs must be sent to some queues oftype r of the ( p r m + i )-th system in the time interval [ V + a ( p r ( m − i ) j,r , V + a ( p r m + i ) j,r − j = 1, andthe number of arrivals at the dispatcher in that interval is exactly (cid:107) p (cid:107) by construction of the policies in C ( k ) p for any k (see subsection 3.1)This argument applies to all the other queues because we have considered periodic policies.We show Fact 1 in the following example, to help understanding its meaning and proof.10 xample 1. Assume R = 2 , p = (3 , , r = 1 , i = 1 , m = 2 , j = 1 . Assume also that π ∈ C (1) p is such that ( π ( V + n )) n =0 ,..., = (1 , , , , . Then, the sequence of queues to be visited for both systems p r m + i and p r ( m −
1) + i is given in Table 1, where we can see that the decomposition in Fact 1 holds. a ( p r m + i ) j,r (cid:122) (cid:125)(cid:124) (cid:123) p r m + i : (1,1) (1,2) (2,1) (1,3) (2,2) (1,4) (1,5) (2,3) (1,6) (2,4) (1,7) (1,1) . . . (cid:124) (cid:123)(cid:122) (cid:125) (cid:107) p (cid:107) p r ( m −
1) + i : (1,1) (1,2) (2,1) (1,3) (2,2) (1,4) (1,1) (2,3) (1,2) (2,4) (1,3) (1,4) . . . (cid:124) (cid:123)(cid:122) (cid:125) a ( p r ( m − i ) j,r Table 1: Illustrative example for the decomposition in Fact 1.Since Fact 1 holds for any m >
1, we can make the replacement m → m lcm ( p ) p r for which we obtain a ( k m + i ) j,r = a ( k m − p r + i ) j,r + (cid:107) p (cid:107) . (30)Unfolding this recursion, for m ∈ N , i = 1 , . . . , lcm( p ), we get a ( k m + i ) j,r = a ( k m − p r + i ) j,r + 2 (cid:107) p (cid:107) = · · · = a ( k m − + i ) j,r + lcm ( p ) p r (cid:107) p (cid:107) = a ( i ) j,r + m (cid:107) p (cid:107) p r lcm( p ) (31)and therefore a ( k m + i ) j,r k m + i = a ( i ) j,r + m (cid:107) p (cid:107) p r lcm( p ) m · lcm( p ) + i −−−−→ m →∞ (cid:107) p (cid:107) p r . (32)As a technical observation, in (31) we note why we require index i to range in { , . . . , lcm( p ) } : it ‘closes’the recursion.Since (32) holds for all i = 1 , . . . , lcm( p ), by using Lemma 1 we obtainlim k →∞ a ( k ) j,r k = (cid:107) p (cid:107) p r . (33)As a comment, we note here that (33) could be proven without Fact 1 and what has followed. However,we stress that we will need Fact 1 later anyhow, as it will play a crucial role in the proof of our main resultTheorem 3 (see Lemma 3.iii). Lemma 2.
Under the hypotheses of Theorem 2, T ( k ) n,r,κ → (cid:107) p (cid:107) λp r in probability, as k → ∞ .Proof. For all (cid:15) > (cid:16) | T ( k ) n,r,κ − (cid:107) p (cid:107) λp r | ≥ (cid:15) (cid:17) = Pr (cid:16) | T ( k ) n,r,κ − E T ( k ) n,r,κ | ≥ (cid:15) (cid:17) (34a) ≤ (cid:15) Var T ( k ) n,r,κ (34b)= 1 (cid:15) (cid:16) E (Var T ( k ) n,r,κ | U r,κ ) + Var E ( T ( k ) n,r,κ | U r,κ ) (cid:17) (34c)= 1 (cid:15) p r p r (cid:88) j =1 Var A ( k ) j,r,κ + ( E A ( k ) j,r,κ − E T ( k ) n,r,κ ) (34d)= 1 (cid:15) p r p r (cid:88) j =1 a ( k ) j,r Var T ( k )1 + (cid:16) a ( k ) j,r λk − (cid:107) p (cid:107) λp r (cid:17) (34e)11 −−−→ k →∞ . (34f)In (34b), (34c) and (34e), we have used Chebyshev’s inequality, the law of total variance and that the T ( k ) n ’sare i.i.d., respectively. In (34f), we have used (33) and that a ( k ) j,r Var T ( k )1 → T ( k )1 = o ( k ) and a ( k ) j,r ≤ k (cid:107) p (cid:107) .Since T ( k ) n,r,κ converges in probability for each n , also the finite dimensional distributions of the process( T ( k ) n,r,κ ) n ∈ N converge to the one of the constant process with rate λp r (cid:107) p (cid:107) . Together with the fact that E T ( k ) n,r,κ =lim k →∞ E T ( k ) n,r,κ = (cid:107) p (cid:107) λp r , we can use the continuity of the stationary waiting time (see [12, Theorem 22]) toestablish that W ( k ) r,κ ( π ( k ) ) d −−−−→ k →∞ W r ( p ) . (35)Using (35) and that Cesaro sums converge if each addend converges, we obtainlim k →∞ Pr( W ( k ) ( π ( k ) ) ≤ t ) = lim k →∞ R (cid:88) r =1 p r (cid:107) p (cid:107) k k (cid:88) κ =1 Pr( W ( k ) r,κ ( π ( k ) ) ≤ t ) (36a)= R (cid:88) r =1 p r (cid:107) p (cid:107) Pr( W r ( p ) ≤ t ) = Pr( W ( p ) ≤ t ) (36b)as desired. Proof of (15) and (16). Given that C ( k ) p ⊆ A p ( k ), (15) and (16) hold true if E W ( p ) ≤ E W ( k ) ( π ) (37)for all π ∈ A p ( k ) and lim k →∞ E W ( k ) ( π ( k ) ) ≤ E W ( p ) . (38)Let E W r,κ ( x ) be the mean waiting time of a D/GI/1 queue with arrival rate λx and i.i.d. service timeshaving the same distribution of S ,r,κ . Inequality (37) is a fairly direct application of known results: for all π ∈ A p ( k ), E W ( k ) ( π ) = (cid:88) r,κ q r,κ E W ( k ) r,κ ( π ) (39a) ≥ (cid:88) r,κ q r,κ E W r,κ ( kq r,κ ) (39b) ≥ (cid:88) r,κ p r k (cid:107) p (cid:107) E W r,κ ( p r (cid:107) p (cid:107) ) (39c)= (cid:88) r p r (cid:107) p (cid:107) E W r ( p ) = E W ( p ) . (39d)In (39b), we have used the lower bound in [23]. In (39c), we have used Karamata’s inequality once noticingthat i) E W r,κ ( x ) = E W r, ( x ), ii) the majorization ( p r (cid:107) p (cid:107) , . . . , p r (cid:107) p (cid:107) ) ≺ ( kq r, , . . . , kq r,k ) holds, and iii) the meanwaiting time of a D/GI/1 queue is convex increasing in the arrival rate (see, e.g., [30, Theorem 5], [17]),which means that q r,κ E W r,κ ( kq r,κ ) is convex in q r,κ .We now prove (38). As in the proof of Theorem 2, this can be done assuming that (28) holds. Thus, thesequences (28) will be assumed along this proof.The remainder of the proof basically works as follows. First, we bound the waiting times of our G/GI/1queues through the waiting times of suitable GI/GI/1 queues. Second, we show that the sequence of such12aiting times converges in distribution to W ( p ). Then, we show that the waiting times of such GI/GI/1queues are non-increasing in the ≤ icx -sense along the sequences k m + i , for all i = 1 , . . . , lcm( p ), which allowsus to conclude that the sequence is uniformly integrable; this is the point where we will use Fact 1. Finally,we use [11, Theorem 3.5, pp. 31] to conclude that also the sequence of the expected values converges to E W ( p ).Associated to each queue of type r , we define an auxiliary random variable, T ( k ) r , such that T ( k ) r = st min j =1 ,...,pr a ( k ) j,r (cid:88) n =1 T ( k ) n . (40)Next lemma provides properties satisfied by T ( k ) r that will be used later. We recall that k m = m lcm( p ),see (29). Lemma 3.
Under the hypotheses of Theorem 3, the following properties hold:i) We have lim k →∞ E T ( k ) r = lim k →∞ min j =1 ,...,p r a ( k ) j,r λk = (cid:107) p (cid:107) λp r . (41) ii) T ( k ) r → (cid:107) p (cid:107) λp r in probability, as k → ∞ .iii) For all i = 1 , . . . , lcm( p ) , − T ( k m +1 + i ) r ≤ icx − T ( k m + i ) r . (42) Proof. Proof of i) . This is an immediate consequence of (33).
Proof of ii) . For all i = 1 , . . . , lcm( p ), let j ∗ i ∈ arg min j =1 ,...,p r a ( i ) j,r i . Then, from (32), we get j ∗ i ∈ arg min j =1 ,...,p r a ( k m + i ) j,r k m + i , ∀ m > . (43)In view of Lemma 1, the convergence in ii) holds if we can show that T ( k m + i ) r Pr −−−−→ m →∞ (cid:107) p (cid:107) λp r , for all i = 1 , . . . , lcm( p ). Given (43), this amounts to show that a ( km + i ) j ∗ i ,r (cid:88) n =1 T ( k m + i ) n Pr −−−−→ m →∞ (cid:107) p (cid:107) λp r , ∀ i = 1 , . . . , lcm( p ) . (44)We prove the former by showing (the stronger statement) that (cid:80) a ( k ) j,r n =1 T ( k ) n Pr −−−−→ k →∞ (cid:107) p (cid:107) λp r , for all j . Now, usingthat {| X − c | > (cid:15) } ⊆ {| X − E X | > (cid:15) } ∪ {| E X − c | > (cid:15) } for a random variable X , we havePr (cid:16) | a ( k ) j,r (cid:88) n =1 T ( k ) n − (cid:107) p (cid:107) p r λ | ≥ (cid:15) (cid:17) ≤ Pr (cid:16) | a ( k ) j,r (cid:88) n =1 T ( k ) n − E a ( k ) j,r (cid:88) n =1 T ( k ) n | ≥ (cid:15) (cid:17) +Pr (cid:16) | (cid:107) p (cid:107) p r λ − E a ( k ) j,r (cid:88) n =1 T ( k ) n | ≥ (cid:15) (cid:17) . The second term in the right-hand side of former inequality tends to zero as k → ∞ by (33). The following13hows that also the first term goes to zero:Pr (cid:16) | a ( k ) j,r (cid:88) n =1 T ( k ) n − E a ( k ) j,r (cid:88) n =1 T ( k ) n | ≥ (cid:15) (cid:17) ≤ (cid:15) a ( k ) j,r (cid:88) n =1 Var T ( k ) n (46a) ≤ (cid:15) a ( k ) j,r o (1 /k ) −−−−→ k →∞ . (46b)In (46a), we have used Chebyshev’s inequality and that the T ( k ) n ’s are independent. In (46b), we have usedthat a ( k ) j,r ≤ k (cid:107) p (cid:107) . Proof of iii) . We use that X ≤ icx Y if and only if there exists an other random variable Z such that X ≤ st Z and Z ≤ cx Y [29].Using (31) and that a ( km + i ) j ∗ i ,r ( k m + i ) ≤ (cid:107) p (cid:107) p r for all m (by (32)), the first observation is that a ( k m +1 + i ) j ∗ i ,r ≥ a ( k m + i ) j ∗ i ,r + lcm( p ) a ( k m + i ) j ∗ i ,r k m + i , (47)where j ∗ i is defined above in point i). Thus, we have − T ( k m +1 + i ) r = st − a ( km +1+ i ) j ∗ i ,r (cid:88) n =1 T ( k m +1 + i ) n (48a) ≤ st − a ( km + i ) j ∗ i ,r + lcm ( p ) a ( km + i ) j ∗ i ,rkm + i (cid:88) n =1 T ( k m +1 + i ) n def = − Z. (48b)Now, it remains to show that − Z ≤ cx − T ( k m + i ) r , which is equivalent to show that Z ≤ cx T ( k m + i ) r , (49)see [33, Theorem 3.A.12]. Since E Z = 1 λ a ( k m + i ) j ∗ i ,r + lcm( p ) a ( km + i ) j ∗ i ,r k m + i k m + i + lcm( p ) = 1 λ a ( k m + i ) j ∗ i ,r k m + i = E T ( k m + i ) r , (50)(49) holds trivially under Case 3. Now, let Case 2 hold. Noticing that both T ( k m + i ) r and Z have Erlangdistributions with the same mean, to prove (49) is enough to show that Var Z ≤ Var T ( k m + i ) r ; see [35, p. 14].We have λ Var Z = a ( k m + i ) j ∗ i ,r + lcm( p ) a ( km + i ) j ∗ i ,r k m + i ( k m +1 + i ) (51a)= a ( k m + i ) j ∗ i ,r ( k m + i ) ( k m + i )( k m + i + lcm( p ))( k m + i + lcm( p )) (51b) ≤ a ( k m + i ) j ∗ i ,r ( k m + i ) = λ Var T ( k m + i ) r (51c)as desired. 14e now present an argument that allows us to uniformly bound the second moment of W ( k ) r,κ .Let δ r def = (cid:16) (cid:107) p (cid:107) p r λ − µ r (cid:17) and k ∗ def = min (cid:26) k > ≤ (cid:107) p (cid:107) p r λ − min j =1 ,...,p r a ( k (cid:48) ) j,r k (cid:48) λ ≤ δ r , ∀ k (cid:48) ≥ k (cid:27) . (52) Lemma 4. k ∗ < ∞ .Proof. This is immediate because δ r > k (cid:48) →∞ min j =1 ,...,p r a ( k (cid:48) ) j,r k (cid:48) λ = (cid:107) p (cid:107) p r λ (see (41)), and (cid:107) p (cid:107) p r λ ≥ min j =1 ,...,p r a ( k ) j,r kλ for all k .Let W ( k ) r denote the stationary waiting time of a GI/GI/1 queue with (i.i.d.) interarrival times ( T ( k ) n,r ) n ∈ N where T ( k ) n,r = st T ( k ) r (see (40)) and service times ( S ( k ) n,κ,r ) n ∈ N . By coupling the T ( k ) n,r ’s and the T ( k ) n,κ,r ’s inthe obvious manner, one can easily see that ( T ( k )1 ,r , . . . , T ( k ) n,r ) ≤ ( T ( k )1 ,κ,r , . . . , T ( k ) n,κ,r ) and therefore we have( T ( k )1 ,r , . . . , T ( k ) n,r ) ≤ st ( T ( k )1 ,κ,r , . . . , T ( k ) n,κ,r ). Using, e.g., [6, pp. 217, 220], this implies W ( k ) r,κ ≤ st W ( k ) r . (53)Furthermore, given that − T ( k m +1 + i ) n,r ≤ icx − T ( k m + i ) n,r for all i = 1 , . . . , lcm( p ) (by Lemma 3) and that the( T ( k ) n,r ) n ∈ N are independent, we can use [5, p. 337] to establish that W ( k m +1 + i ) r ≤ icx W ( k m + i ) r , (54)for all i = 1 , . . . , lcm( p ). Therefore, given E (cid:16) ( W ( k ) r,κ ) (cid:17) < ∞ for all k and Lemma 4, we can uniformly boundthe second moment of W ( k ) r,κ as followssup k ≥ k ∗ E (cid:16) ( W ( k ) r,κ ) (cid:17) ≤ sup k ≥ k ∗ E (cid:16) ( W ( k ) r ) (cid:17) (55a)= max i =1 ,..., lcm ( p ) sup m : k m + i ≥ k ∗ E (cid:16) ( W ( k m + i ) r ) (cid:17) (55b)= max i =1 ,..., lcm ( p ) E (cid:16) ( W ( k m ∗ i + i ) r ) (cid:17) (55c) < ∞ , (55d)where m ∗ i def = min { m : k m + i ≥ k ∗ } . In (55a) and (55c), we have used (53) and (54), respectively. In (55d),we have used that E T ( k ) n,r = min j =1 ,...,p r a ( k ) j,r kλ ≥ (cid:107) p (cid:107) p r λ − δ r = (cid:107) p (cid:107) p r λ +
12 1 µ r > µ r , ∀ k ≥ k ∗ , (56)i.e. the ergodicity condition, and that the third moment of service times is finite, which imply that thesecond moment of W ( k ) r is finite [5, pg. 270].Now, using the continuity of the stationary waiting time of GI/GI/1 queues [5, Corollary X.6.4] and parti) and ii) of Lemma 3, we have W ( k ) r d −−−−→ k →∞ W r ( p ) , and given the uniform integrability (55) we have thatalso the expected values converge [11, Theorem 3.5, pp. 31], i.e.,lim k →∞ E W ( k ) r = E W r ( p ) . (57) Given x, y ∈ R d , here x ≤ y means x i ≤ y i for all i = 1 , . . . , d . k →∞ E W ( k ) ( π ( k ) ) = lim k →∞ (cid:88) r (cid:88) κ p r (cid:107) p (cid:107) k E W ( k ) r,κ ( π ( k ) ) (58a) ≤ lim k →∞ (cid:88) r p r (cid:107) p (cid:107) E W ( k ) r (58b)= (cid:88) r p r (cid:107) p (cid:107) E W r ( p ) . (58c) Proof of (17). Let Q be a discrete random variable with values in { , . . . , R } × { , . . . , k } such that Pr( Q =( r, κ )) = p r (cid:107) p (cid:107) k . We assume that this random variable is independent of any other random variable. Bydefinition of W ( k ) ( π ( k ) ) and using the law of total variance, we obtainVar W ( k ) ( π ( k ) ) = E (Var W ( k ) ( π ( k ) ) | Q ) + Var E ( W ( k ) ( π ( k ) ) | Q ) (59a)= (cid:88) r,κ p r (cid:107) p (cid:107) k (cid:18) Var W ( k ) r,κ ( π ( k ) ) + (cid:16) E W ( k ) r,κ ( π ( k ) ) − E W ( k ) ( π ( k ) ) (cid:17) (cid:19) . (59b)When k → ∞ , we have already established that E W ( k ) ( π ( k ) ) → E W ( p ) and that E W r ( p ) ≤ E W ( k ) r,κ ( π ( k ) ) ≤ E W ( k ) r,κ → E W r ( p ). Therefore, it only remains to show that the second moment of W ( k ) r,κ ( π ( k ) ) converge to E [ W r ( p ) ]. This is done by using the same argument above for the convergence of the first moment. Hence,using the continuity of the waiting time and of the square function [5, Corollary X.6.4] and part i) and ii) ofLemma 3, we obtain ( W ( k ) r ) d −→ W r ( p ) , as k → ∞ . Furthermore, the second moment of ( W ( k ) r ) is finitebecause the fifth moment of the service times is finite [5, pg. 270] and (54) ensures that the sequence ( W ( k ) r ) is uniformly integrable because it is non-increasing along subsequences ( k m + i ) m ∈ N , for all i = 1 , . . . , lcm( p ).Thus, E [( W ( k ) r ) ] → E [ W r ( p ) ]. Together with (18), as desired we obtainlim k →∞ E [( W ( k ) r,κ ( π ( k ) )) ] = E [ W r ( p ) ] . (60) We have derived structural properties concerning a known problem in the literature of stochastic scheduling,that is Problem 1. Fixing the proportion of jobs to send on each queue, p , we have identified a class ofperiodic policies and have proven that all the policies in this class are asymptotically equivalent and optimal.The limiting mean waiting time achieved by these policies, E W ( p ) (see (8)), is expressed in terms of a linearcombination of independent D/GI/1 queues and has the convenient property of being convex in p . We believethat these structural properties provide researchers and practitioners with new means about the consideredproblem. For instance, one consequence of these results is that the problem of computing the optimalproportions of jobs to send to each queue, which is considered a difficult problem (see the introduction),boils down, asymptotically, to the solution of an optimization problem of the form:min E W ( p ) s.t.: p ∈ S , for S compact and convex, and we stress that E W ( p ) is a convex function of p . Using a classic result inconvex optimization, this means that a polynomial number of evaluations of the objective function E W ( p )are sufficient to converge to an optimizer of the problem. Given that each objective evaluation is efficient[30, 34, 32, 13, 14, 8], this lets us conclude that we have significantly reduced much of the difficulty ofProblem 1. In the case where service times have an exponential distribution, E W ( p ) admits a very simplecharacterization because it is the weighted mean waiting time of R D/M/1 queues [9, 4].16 eferences [1] E. Altman, B. Gaujal, and A. Hordijk. Balanced sequences and optimal routing.
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