Control of parallel non-observable queues: asymptotic equivalence and optimality of periodic policies

We consider a queueing system composed of a dispatcher that routes deterministically jobs to a set of non-observable queues working in parallel. In this setting, the fundamental problem is which policy should the dispatcher implement to minimize the stationary mean waiting time of the incoming jobs. We present a structural property 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 $k \sum_r p_r$ and such that exactly $p_r$ jobs are sent in a period to each queue of type $r$. When $k\to\infty$, our main result shows 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 other policy having the same aggregate arrival rate to \emph{all} queues of a given type can do better in minimizing the stationary mean waiting time. This property holds in a strong probabilistic sense. Furthermore, the limiting mean waiting time achieved by our policies is a convex function of the arrival rate in each queue, which facilitates the development of a further optimization aimed at solving the fundamental problem above for large systems.


Introduction
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 dispatcher implement to minimize the mean delay experienced by jobs.There is a vast literature on this subject and the 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 scenario where: • 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-arrival times 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 stationary delay and of determining the minimum mean stationary delay are both considered difficult; see, e.g., [2,1] for an overview.A policy can be defined as a function that maps a natural number n, corresponding to the n-th In this paper, we focus on the first subproblem and, under some system scaling, we identify a set of policies that 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 minimizes the waiting time of the single queue [18,23].In view of this classic insight and fixing fractions p, it is not surprising that an optimal policy tries to make the arrival process at each resource as regular (or less variable) as possible.Thus, our stochastic scheduling problem can be essentially converted into a problem in word combinatorics.If the dispatcher must ensure fractions p, the main result known in the literature is that 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, which is still open to the best of our knowledge [36]; see also [2, Chapter 2], which contains an overview of which rates 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-robin1 is known to be optimal in a strong sense [27], or when the dispatcher routes jobs to two resources, where balanced sequences can be always constructed no matter the value of rates p [19].In presence of more than two queues, we stress that balanced sequences with given rates p do not exist in general.This non-existence makes 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 a Poisson process and service times have an exponential distribution, the optimal rates p as function of the inter-arrival and service times have a fractal structure, see [16,Figure 8].This puts further light on the complexity of the problem even in a simple scenario.
While deterministic policies are believed to be more difficult to study than probabilistic policies (deterministic 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 each resource is much more regular in the deterministic case, especially if there are several resources as we show in this paper.A particular class of deterministic policies, namely billiard policies, have been recently implemented in the context of large volunteer and cloud computing to improve the performance of real applications such as SETI@home [24,25].

Contribution
In the framework described above, we are interested in deriving structural properties of deterministic policies when the system size is large.We study a scaling of the system where the arrival rate of jobs, λk, grows to infinity proportionally with the number of resources (queues in the following), Rk, while keeping the network load (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 to infinity.Beyond issues related to the tractability of the problem, this type of scaling is motivated by the fact that the size of real systems is large and that replication of resources is commonly used to increase system reliability.
First, with respect to a class of periodic policies, we define the random variable of the waiting time of each incoming job.This is done using Lindley's equation [26] and a suitable initial randomization.Using such randomization, we can adapt the framework developed by Loynes in [28] to our setting where jobs are sent to a set of parallel queues.In particular, Theorem 1 shows the monotone convergence in distribution of the waiting time of each incoming job.Then, with respect to a given vector p ∈ N R , we define a certain subset of policies that are periodic with period k r p r and such that exactly p r jobs are sent in a period to each queue of type r.While further details will be developed in Section 3.1, this set is meant to imply that queues of a given type are visited in a round-robin manner and that arrivals are "well distributed" among the 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 other policy having the same aggregate arrival rate to all queues of a given type can do better in minimizing the mean stationary waiting time.In particular, we show that the stationary waiting time converges both in distribution and in expectation to the stationary waiting time of a system of independent D/GI/1 queues whose parameters only depend on p, λ and the distribution of the service times.This is shown in Theorems 2 and 3, respectively.
The main idea underlying our proof stands in analyzing the sequence of stationary waiting times along appropriate subsequences.Along these subsequences, it is possible to extract a pattern for the arrival process of each queue that is common to all members of the subsequence.Such pattern is exploited to establish monotonicity properties in the language of stochastic orderings.These properties hold for the considered subsequences only: they do not hold true along any arbitrary subsequence and counterexamples can be given.These properties will imply the uniform integrability of the sequence of stationary waiting times and will 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 results state that all the policies belonging to the set that we identify in Section 3.1 yield the same asymptotic performance and are asymptotically optimal.Furthermore, using known properties of the D/GI/1 queue, we obtain that the stationary mean waiting time obtained in our limit is a convex function of p.This reduces the complexity 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 a characterization of the stationary waiting time (Theorem 1); Section 3 introduces a class of policies and presents our main results (Theorems 2 and 3); Section 4 is devoted to proofs; finally, Section 5 draws the conclusions of this paper.

Parallel queueing model
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-comefirst-served (FCFS) and the buffer size of each queue is infinite.A stream of jobs (or customers) joins the queues through a dispatcher.The dispatcher routes each incoming job to a queue according to some policy and instantaneously.Figure 1 illustrates the structure of the queueing model under investigation.In the following, 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 probability triple (Ω, F, Pr).
Let (T n,κ,r ) n∈N be given sequences of i.i.d.random variables in R +2 .These sequences are all assumed to be independent each other.Quantity T (k) n is interpreted as the inter-arrival time between the n-th and the (n + 1)-th jobs arriving to the dispatcher.Quantity S (k) n,κ,r is interpreted as the service times of the n-th job arriving at the κ-th queue of type r.We assume that S 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 It is clear that Cases 2 and 3 are both more restrictive than Case 1.Let • denote the L 1 -norm. Let Quantity q r,κ will be interpreted as the proportion of jobs sent to queue (r, κ).
Since q is a vector of rational numbers, n * < ∞.Let V be a discrete random variable with values in {1, . . ., 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 → {1, . . ., R} × {1, . . ., k} such that for all r and κ q for all n, where 1 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) means that the n-th job arriving to the dispatcher is sent to the k-th queue of type r if n ≥ V , otherwise it means that the n-th job is discarded.Thus, the outcome of random variable V gives the index of the first job that is 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 queue of type r (under any of the cases above).By construction, T (k) n,κ,r (π) is the sum of a deterministic number of inter-arrival times seen at the dispatcher.The arrival process (T (k) n,κ,r (π)) n∈N can be made stationary if it is allowed a shift in time and a suitable randomization (independent of V and of any other random variable) for the inter-arrival time of the first arrival of each queue.We assume for now that this has been done (details will be given at the beginning of Section 4).As done in [28] and according to [15, p. 456], this implies that we 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, the same holds for the process (S The waiting time of the n-th job arriving to the κ-th queue of type r induced by a policy π ∈ A(k) is denoted by W (k) n,r,κ (π).It is the time between its arrival at the dispatcher (or equivalently at the queue) and the start of its service, and it is defined as follows: for n = 0, W (k) n,r,κ (π) = 0 and for n > 0, where x + def = max{x, 0}.Equation ( 2) is known as Lindley's recursion [26].The assumption that W (k) 0,r,κ (π) = 0 serves to avoid technicalities3 .It is known that the sequence of random variables (W and that 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×{1, . . ., R}×{1, . . ., k} be a mapping with the following meaning: f (n) = (n , r, κ) means that the n-th job arriving to the dispatcher is the n -th customer joining queue (r, κ).If n < V , no job is sent to any queue and thus we assume where f j refers to the j-th component of f .With this notation, quantity 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 {1, . . ., n * }.The subsets Q r,κ , for all r and k, are thus disjoint and we further require that the number of points in Q r,κ is n * q r,κ .Let r,κ (π)'s with weights q.Next theorem says that W (k) (π) can be interpreted as the right random variable describing the stationary waiting time of jobs achieved with policy π ∈ A q (k).It is proven by adapting the framework developed by Loynes in [28].In the remainder of the paper, convergence in distribution (in probability) is denoted by 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, and By using the monotone convergence theorem, Theorem 1 implies that also the moments of Finally, for a given p ∈ R R + , we define the auxiliary random variable W r (p), which corresponds to the stationary waiting time of a D/GI/1 queue with inter-arrival times (T n,r (p)) n∈N where T n,r (p) = p /(p r λ) and service times (S n,1,r ) n∈N , for all r.Let also V def = V (p) be a random variable with values in {1, . . ., R} independent of any other random variable and such that Pr(V = r) = p r / p , for all r, and let Note that EW (p) = r pr p EW r (p) can be interpreted as the mean waiting time of R independent D/GI/1 queues averaged over weights p/ p .
We will be interested in establishing convergence results for W (k) when k → ∞.With respect to some policies to be defined, we will show forms of convergence to W (p).

Discussion
Some remarks about the model above and Theorem 1 follow: • To agree on a definition for EW (k) or other moments of W (k) , one does not necessarily need to know the distribution of W (k) , e.g., one can achieve that using Cesaro sums [2].Matter of fact, existing works agree on the structure of EW (k) without constructing the distribution of W (k) .On the other hand, our approach needs to know the distribution of W (k) (and thus Theorem 1) because we can prove convergence results for EW (k) only through a distributional convergence argument, that is [11,Theorem 3.5,pp. 31].In particular, to prove EW (k) converges to EW (p), we will prove convergence in distribution of W (k) to W (p) and then the uniform integrability of the sequence of the W (k) 's.This is the 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 EW (k) (see the introduction), the analysis of EW (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 of generality for obvious reasons.As an additional remark to support this assumption, we note that it has been proven that in several cases the q-vector that minimize min π∈Aq(k) EW (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 case that we do not consider here, and the limit lim n→∞ does not exist, it would be interesting to know whether some convergence in distribution of W (k) f (n) (π) occurs.A totally different argument 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, as we do not let k → ∞ there.Case 1 covers the case where T (k) n has a (Gk, α)-phase-type distribution where 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 though we do not do it for simplicity.

Main results
For p ∈ Z R + , let and The set A p (k) is interpreted as the set of all periodic policies for which λk pr p is the aggregate mean arrival rate of jobs to all type-r queues.We consider the following problem.
As discussed in the introduction, this is considered as a difficult problem.We are interested in establishing structural properties of Problem 1 when k is large.In the following, we define a certain class of policies and then present our main results.

A class of periodic policies
as the subset of all policies π ∈ A p (k) that satisfy the following properties: • The sequence (π 1 (n)) n∈N has period p and p r is the number of jobs sent to type-r queues per period, for 1 ≤ r ≤ R.
The first property specifies the periodicity of π with respect the types of queues, while the second with respect 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 p |, does not vary with k.One can verify that These policies can be implemented in a distributed manner.More precisely, one can think that there are two tiers of dispatchers: the dispatcher in the first tier schedules jobs inter -group, while the dispatchers in the 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 p , we will show (Lemma 2) This means that the finite dimensional distributions of the arrival process at each queue will be 'close' to the deterministic process, when k is large, which implies that some form of convergence to W (p) should occur in view of the continuity of the stationary waiting time [12].

Asymptotic equivalence and optimality
Next theorem proves a first form of convergence of Theorem 2. Let Case 1 hold.Let p ∈ Z R + be such that λ pr p < µ r for all r.Let also an arbitrary sequence (π (k) ) k∈N be given where π Theorem 2 is not enough to claim that EW (k) (π (k) ) converges as well.Convergence of the first moment is important from an operational standpoint, as in practice one desires to optimize over EW (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 and asymptotically optimal, with respect to the criterion in Problem 1.
Theorem 3. Let Case 2 or 3 hold.Let p ∈ Z R + be such that λ pr p < µ r for all r.Let also an arbitrary sequence (π (k) ) k∈N be given where π 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 to minimize EW (k) .It also explicits the limiting value of EW (k) (π (k) ), which is EW (p).It is known that EW (p) is a convex function in p (e.g., [30]).
To prove Theorem 3, we use the well-known fact that [5] 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 its arrival rate, it is not difficult to show that Then, we prove that the sequence EW (k) (π (k) ) is upper bounded by a sequence that converges to the lower bound in (19).An observation here is that the lower bound (18) holds under conditions that are weaker than those assumed in this paper; see [23].For instance, it is possible to extend (19) (and thus Theorem 3) to the case where i) policies are not periodic, ii) the fractions of jobs to send in each queue are not necessarily rational numbers, and iii) policies are randomized [31], that is the case where π(n) is any probability mass function over the set of queues.We do not investigate these extensions in further detail.

Proofs
In this section, we develop proofs for Theorems 1, 2 and 3. Before doing this, we fix some additional notation and show how it is possible to make the arrival process at each queue stationary with respect to any policy in π (k) ∈ A q (k), q ∈ Q kR + such that q = 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 clearly depends on the policy π (k) implemented by the dispatcher, i.e., T n,κ,r (π (k) ), though in the following we 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 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, where = st denotes equality in distribution. If for some p ∈ Z R + , then we notice that a (k) j,r,κ does not vary with κ because by symmetry the arrival processes of all queues of a given type are equal, in distribution, up to a shift in time.In this case, 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 2,κ,r , . . ., A (k)  pr,κ,r .
Therefore, when p , we will just write a and that Now, we want to make (T n,κ,r ) n∈N stationary.This can be done as follows by randomizing over the first 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 uniform distribution in [0, 1].Then, we take Therefore, if j+1,r,κ and so forth according to the pattern (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 , κ ), and suppose that T j ,r ,κ .We should check whether policy π (k) is actually able to induce arrival processes at the queues equal (samplepath-wise) to the ones built above through (25).One can easily see that this can be done with a possible shift of time for the arrival process at the queues and possibly discarding a finite number of 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 usual stochastic 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×{1,...,N } is a semi-infinite array of numbers such that for some constant c, lim k→∞ f k,n = c, for n ∈ {1, . . ., N }.Then, for any sequence We now give proofs for our results, i.e., Theorems 1, 2 and 3.

Proof of Theorem 1
Let random variables V 1 and V 2 be given such that f (n+1) (π) be the Loynes waiting times (see [28, p. 501]) obtained when the first queue to serve a job is given by the outcome of V 1 and V 2 , respectively; thus, W (k) f (n) (π) is the waiting time at time 0 with n jobs in the past at queue (r, κ), provided that f (n) = (n , r, κ).Then, we let the (Loynes) waiting times be driven by the same realizations of the random inter-arrival and service times.
We now prove (7).Since π is periodic with period n * , we first observe that π returns the same queue along 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 they depend on the outcome of random variable V , see (4).Thus, for all i = n * + 1, . . ., 2n * , by construction we have Pr(f and we get In (27b), we have conditioned on f (i).In (27c), we have used that r,κ (π); see [28].In (27d), we have used the definition of W (k) (π).Now, since the limit in (27d) does not depend on i, the proof is concluded by applying Lemma 1 once noted that n * is a finite positive integer.

Proof of Theorem 2
We first observe that we can prove this theorem under some assumption on the sequence (π (k) ) k∈N .Given π (1) ∈ C p , we require that for all k: One may refer to these sequences as the 'natural' scaling of policy π (1) : in the two-tier interpretation of our policies, (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 also holds for all the sequences in view of Lemma 1 and of the fact that the cardinality of C (V ) = 1 and π (prm+i) (V ) = π (pr(m−1)+i) (V ), which in some sense couples 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 of type r of the (p r m + i)-th system in the time interval [V + a (pr(m−1)+i) j,r , V + a 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.
Since (32) holds for all i = 1, . . ., lcm(p), by using Lemma 1 we obtain 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 result Theorem 3 (see Lemma 3.iii).
Lemma 2. Under the hypotheses of Theorem 2, In (34b), (34c) and (34e), we have used Chebyshev's inequality, the law of total variance and that the T (k) n 's are i.i.d., respectively.In (34f), we have used (33) and that a n,r,κ = p λpr , we can use the continuity of the stationary waiting time (see [12,Theorem 22]) to establish that Using (35) and that Cesaro sums converge if each addend converges, we obtain as desired.

Proof of Theorem 3
Proof of (15) and ( 16).Given that C p ⊆ A p (k), ( 15) and ( 16) hold true if for all π ∈ A p (k) and lim k→∞ Let EW r,κ (x) be the mean waiting time of a D/GI/1 queue with arrival rate λx and i.i.d.service times having the same distribution of S 1,r,κ .Inequality (37) is a fairly direct application of known results: for all π ∈ A p (k), In (39b), we have used the lower bound in [23].In (39c), we have used Karamata's inequality once noticing that i) EW r,κ (x) = EW r,1 (x), ii) the majorization ( pr p , . . ., pr p ) ≺ (kq r,1 , . . ., kq r,k ) holds, and iii) the mean waiting 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 EW 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, the sequences (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/1 queues through the waiting times of suitable GI/GI/1 queues.Second, we show that the sequence of such waiting times converges in distribution to W (p).Then, we show that the waiting times of such GI/GI/1 queues are non-increasing in the ≤ icx -sense along the sequences k m + i, for all i = 1, . . ., lcm(p), which allows us 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 EW (p).
Associated to each queue of type r, we define an auxiliary random variable, T r , such that Next lemma provides properties satisfied by T (k) r that will be used later.We recall that k m = m lcm(p), see (29).iii) For all i = 1, . . ., lcm(p), 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,...,pr a 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) where j * i is defined above in point i).Thus, we have −T

Figure 1 :
Figure 1: Structure of the parallel queueing model under investigation.

Problem 1 .
Let p ∈ Z R + be given.Determine the optimizers and the optimal objective function value of min π∈Ap(k)

Fact 1 .
For m ∈ N, let k m def = m lcm(p),(29)where lcm(p) denotes the least common multiple of p 1 , . . ., 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 true and follows by construction of the policies in set C(k)p : it is a direct consequence of the fact that queues of the same type are visited in a round-robin manner.For m > 1, j = 1, . . ., p r and i ∈ N, a By construction, we have π (prm+i) 2

− 1 ]
, for j = 1, and the number of arrivals at the dispatcher in that interval is exactly p by construction of the policies in C (k) p for any k (see subsection3.1) ,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 λpr p .Together with the fact that ET (k) n,r,κ = lim k→∞ ET (k)

Lemma 3 .
Under the hypotheses of Theorem 3, the following properties hold: λpr in probability, as k → ∞.
Lemma 1, the convergence in ii) holds if we can show that T all i = 1, . . ., lcm(p).Given (43), this amounts to show that a ∀i = 1, . . ., lcm(p).(44)Weprove the former by showing (the stronger statement) that a all j.Now, using that{|X − c| > 2 } ⊆ {|X − EX| > } ∪ {|EX − c| > } for a random variable X,The second term in the right-hand side of former inequality tends to zero as k → ∞ by(33).The following shows that also the first term goes to zero: ), we have used Chebyshev's inequality and that the T (k) n 's are independent.In (46b), we have used that a (k) j,r ≤ k p .

(k m + i) 2 (
remains to show that −Z ≤ cx −T (km+i) r , which is equivalent to show that Z ≤ cx T holds trivially under Case 3. Now, let Case 2 hold.Noticing that both T (km+i) r and Z have Erlang distributions with the same mean, to prove (49) is enough to show that Var Z ≤ Var T k m+ i)(k m + i + lcm(p)) (k m + i + lcm(p)) 2 (51b) ≤ a (km+i) j * i ,r (k m + i) 2 = λ 2 Var T (km+i) r(51c)as desired.

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) and that