Laws of Little in an open queueing network

Abstract. The object of this research in the queueing theory is theorems about the functional strong laws of large numbers (FSLLN) under the conditions of heavy traffic in an open queueing network (OQN). The FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for the values of important probabilistic characteristics of the OQN investigated as well as the virtual waiting time of a customer and the queue length of customers. As applications of the proved theorems laws of Little in OQN are presented.


Introduction
The paper is devoted to the analysis of queueing systems in the context of the network and communications theory.We investigate FSLLN about the virtual waiting time of a customer and the queue length of customers and theorems on the laws of Little in an OQN under the conditions of heavy traffic.Queueing networks have been extensively used for the analysis of manufacturing systems, transportation systems, and computer and communications networks.Therefore, many approximation methods have emerged, and FSLLN are among them.
The investigation of delays arising in communications and computer systems is a very complicated problem which has not yet been solved in the general case.A valuable progress in this area has been achieved for models based on Kleinrock's hypothesis on the independence of transmission times of messages at different nodes (see [1][2][3][4][5][6][7]).The most fruitful approach to the calculation of delays in communication systems is based on limit theorems for heavy traffic and light traffic regimes.Some results have been obtained both for systems with finite (see [6,8]) and infinite (see [2,7,[9][10][11][12][13][14][15][16][17][18][19]) waiting rooms.The theoretical base for heavy traffic limit theorems includes the weak convergence results for stochastic processes (see [20][21][22][23]) as well as the martingale approach to limit theorems (see [24][25][26]).The limit theorems usually state that, in the heavy traffic regime, c Vilnius University, 2012 properly normalized random variables (or random processes) describing queue lengths or waiting times converge in distribution to a normal random variable (or certain diffusion processes).
The first results for OQN in heavy traffic were obtained by Iglehart and Whitt [14,15].They considered a single station, multiserver and acyclic networks of queues.The limit process for networks in heavy traffic was described as a complicated functional of multidimensional Brownian motion.Harrison [10] considered the heavy traffic approximation to the stationary distribution of the waiting times in single server queues in series.His limit process was also given as a complicated functional of Brownian motion.In [11], Harrison again considered the diffusion approximation for tandem queues, described the limit process and found analytical solutions in several special cases.Reiman [27] proved the heavy traffic limit theorems for the queue length process associated with open queueing networks.These theorems state that the limit process is a reflected Brownian motion on the nonnegative orthant with constant directions for each boundary hyperplane.Harrison and Reiman [12] considered the properties of distribution of the multidimensional reflected Brownian motion.Harrison and Williams [13] also analysed Brownian models of open queueing networks with homogeneous customer populations.Reiman [16] studied a multiclass feedback queue in heavy traffic.A network of priority queues with one bottleneck station in heavy traffic was considered by Reiman and Simon [17].One may note that the theory of heavy traffic analysis is rather well developed for systems that satisfy the Kleinrock hypothesis.Without this hypothesis the complexity of the problem dramatically increases.Now let us review the latest achievements in open queueing networks theory after 2000.The book by Chen and Yao [28] contains the essentials of queueing networks from the classical product-form theory to more recent developments such as diffusion and fluid limits, stochastic comparisons, stability, dynamic scheduling, and optimization.The authors Chen and Zhang of paper [29] have established a sufficient condition for the existence of the (conventional) diffusion approximation to multiclass queueing networks under priority service disciplines.In [30], the authors Chen and Ye have extended the work of Chen and Zhang [29] and established a new sufficient condition for the existence of the (conventional) diffusion approximation to multiclass queueing networks under priority service disciplines.This sufficient condition is related to the weak stability of the fluid networks and the stability of high priority classes of fluid networks that correspond to the queueing networks under consideration.Using a slight modification method, the authors Bramson and Dai of [31] prove heavy traffic limit theorems for six families of multiclass queueing networks (for example, the first three families are singlestation systems operating under first-in-first-out (FIFO) principle, generalized-head-ofthe-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines).The paper of Harrison [32] describes a general type of the stochastic system model that involves three basic elements: activities, resources, and stocks of material.The system manager chooses activity levels dynamically based on state observations, consuming some materials as inputs and producing other materials as outputs, subject to resource capacity constraints.In [33], Mandelbaum and Stolyar investigate a queueing system with multitype customers and flexible (multiskilled) servers that work in parallel, the system in heavy traffic is analyzed and it is shown that a very simple generalized cµ-rule minimizes both instantaneous and cumulative queueing costs asymptotically, over essentially all scheduling disciplines, preemptive or non-preemptive.In [34], Kang, Kelly, Lee and Williams consider a connection-level model of Internet congestion control that represents a randomly varying number of flows present in a network.Here bandwidth is fairly shared amongst elastic document transfers according to a weighted bandwidth sharing policy.Ye and Yao in [35] study a stochastic network that consists of a set of servers processing multiple classes of jobs.Each class of jobs requires a concurrent occupancy of several servers while being processed, and each server is shared among the job classes in a head-of-the-line processor-sharing mechanism.
Limit theorems (diffusion approximations) and the FSLLN for the queueing system under the conditions of heavy traffic are closely connected (they belong to the same field of research, i.e., investigations on the theory of queueing systems in heavy traffic).Therefore, first we shall try to trace the development of research on the general theory of a queueing system in heavy traffic.There is a vast literature on the diffusion approximation.Readers are referred to [7,36,37] and [38] for a general survey of the subject.The present work extends the studies of Iglehart and Whitt [14,15] about a single station of multiserver queues, and of Reiman [27], Johnson [39], Chen and Mandelbaum [40] about networks of single server queues.Other closely related papers are by Harrison and Lemoine [41] on networks of infinite server queues, and Whitt [42] on a GI /G/∞ queue.
The natural setting for functional limit theorems in this paper is the weak convergence of probability measures on the function space D[0, 1].Since an excellent treatment of this subject has been recently published by Billingsley [20], we shall only make a few remarks here about our terminology and notation.Stochastic processes characterizing the queueing system give rise to sequences of random functions in D, the space of all rightcontinuous functions on [0, 1] having left limits and endowed with Skorokhod metric, d.In [20], this metric is denoted by d 0 .With d, D is a complete, separable metric space.Let D be the class of Borel sets of D. Then, if P n and P are probability measures on D which satisfy for every bounded, continuous, real-valued function f on D, we say that P n weakly converges to P , as n → ∞, and write P n ⇒ P .A random function X is a measurable mapping from some probability space (Ω, B, P) into D having the distribution P = PX −1 on (D, D).We say that a sequence of random functions {X n } weakly converges to the random function X, and write X n ⇒ X if the distribution P n of X n converges to the distribution P of X.A sequence of random functions {X n } weakly converges to X in probability if X n and X are defined on a common domain and for all ε > 0, P{d(X n , X) ≥ ε} → 0. When X is a constant function (not random), the convergence in probability is equivalent to a weak convergence.In such cases, we write d(X n , X) ⇒ 0 or X n ⇒ X.If X n and Y n have a common domain, we also write d(X n , Y n ) ⇒ 0, P{d(X n , Y n ) > ε} → 0 as for all ε > 0. We also use the uniform metric ρ which is defined by ρ(x, y) = sup 0≤t≤1 |x(t) − y(t)| for x, y ∈ D. Also, note that d(x, y) ≤ ρ(x, y) for x, y ∈ D.
Next, we state two extremely useful theorems for obtaining weak convergence results in applications.The first has come to be known as the "converging together theorem".For it we assume that X n and Y n are defined on a common domain and take values in a separable metric space (S, m).This result can be found in [20,Thm. 4.1].
Now, suppose h is a measurable mapping of S into S , a second metric space with Borel sets B. Each probability measure P on (S, B) induces on (S , B ) a unique probability measure P h −1 (A) = P (h −1 A) for A ∈ B .Let D h be a set of discontinuities of h.The next result, known as the continuous mapping theorem, is an analog of the Mann-Wald theorem for Euclidean spaces (see [20,Thm. 5 In practice we use this result as follows.First we show X n ⇒ X, often by just quoting the known results.Then, we find an appropriate mapping h which gives us the random elements we are really interested in, h • X n , and finally, we apply Theorem 2.
In this paper, we investigate an OQN model in heavy traffic.We present the FSLLN about the virtual waiting time of a customer and the queue length of customers in an OQN.

The network model
First we consider OQN with the "first come, first served" service discipline at each station and general distributions of interarrival and service time.The basic components of the queueing network are arrival processes, service processes, and routing processes.Particularly, there are k nodes with the node j having a single server and a waiting room of unlimited capacity.The external input stream to the node j is a renewal process, the interarrival time of this process with the mean λ j = (E[z (j) n ]) −1 > 0, and finite variance a j = Dz (j) n > 0, j = 1, 2, . . ., k.These external input streams at the various nodes are assumed to be independent.The service times at the node j are independent and have a common distribution with the mean n > 0, j = 1, 2, . . ., k.The service times at the node j are also independent of all customer arrivals at the node j.A customer leaving the node j is immediately and independently routed to the node i with probability p ji ; and the customer departs the system from the node j with probability p j = 1 − k i=1 p ji .The k × k matrix P = (p ij ) is assumed to have a spectral radius strictly smaller than a unit (see [27]).Note that this system is quite general, encompassing the tandem system, acyclic networks of GI /G/1 queues, and networks of GI /G/1 queues with feedback.
In the context of the queueing network, the random variables z (j) n function as interarrival times (from outside the network) at the station j, while S (j) n is the nth service time www.mii.lt/NA at the station j, and Φ (j) n is a routing indicator for the nth customer served at the station j.If Φ (i) n = j (which occurs with probability p ij ), then the nth customer, served at the station i, is routed to the station j.When Φ (i) n = 0, the associated customer leaves the network.The matrix P is called a routing matrix.
To construct renewal processes, generated by the interarrival and service times, we assume z j (0) = 0, m , l ≥ 1, j = 1, 2, . . ., k.We now define a j (t) = max l ≥ 0: z j (l) ≤ t , x j (t) = max l ≥ 0: S j (l) ≤ t and denote by τ j (t) the total number of customer service departure from the jth station of the network until time t, by τj (t) the total number of customer arrival at the jth station of the network until time t, by τ ij (t) the total number of customers after service that depart from the ith station of the network and arrive at the jth station of the network until time t, and by p t ij = τ ij (t)/τ i (t) a part of the total number of customers which, after service at the ith station of the network, arrive at the jth station; i, j = 1, 2, . . ., k and t > 0.
At first, let us define Q j (t) as the queue length of customers at the jth station of the queueing network at time t, n > 0, j = 1, 2, . . ., k. Next, let us denote by W j (t) the virtual waiting time of a customer in the jth station of the queueing network at time t (time one must wait until a customer arrives at the jth station of the queueing network to be served at time t), βj = (λ j + . ., k.Also, let us denote by S j (t) the time, which can be a sum of summary times of served customers, arriving at the jth station until time t; S j (t) = 2, . . ., k and t > 0. In addition, let [x] be the integer part of number x.Note that, if S j (0) = V j (0) = 0 (OQN at time 0 is empty), then W j (t) = f t (y j (•)), j = 1, 2, . . ., k (see, for example, [1, p. 41].
Assume that the following conditions are fulfilled: Note that these conditions guarantee that, with probability one there exists a queue length of customers and this queue length of customers is constantly growing.In addition, we assume throughout that Nonlinear Anal.Model.Control, 2012, Vol.17, No. 3, 327-342 Conditions ( 2) and ( 3) imply the Lindeberg condition for the respective sequences, and are easier to verify in practice (usually ε = 1 works).

On a fluid limit of the queue length of customers in an OQN
Now we prove the following theorem for a fluid limit of the queue length of customers in an OQN.
Applying the results of the lemma in [44] we obtain that the first term in (28) converges to zero.Also, applying the limit theorem for a renewal process, we see that the second and fourth terms in (28) also converge to zero.Thus, applying (4), we achieve that the third term converges to zero.From these notes we obtain that, for ε > 0, lim t→∞ P Q j (t) t − β j > ε = 0, j = 1, 2, . . ., k.
The proof of the theorem is complete.

The laws of Little for extreme values in an OQN
Here we present applications of the proved theorems (the laws of Little).Proof.To this end, we make use of Theorems 4 and 3 on fluid limits for the queue length of customers and a virtual waiting time of a customer in an OQN.