Network Of Single-server Queues Research Articles

Large deviations for acyclic networks of queues with correlated Gaussian inputs

We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the ‘overflow probability’. In particular, we first leverage Schilder’s sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are nonnegatively correlated, non-short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs.

Non-cooperative queueing games on a network of single server queues

This paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player’s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a 2times 2-grid, and (iv) 2-player games on an Atimes B-grid with small differences in the service rates.

Multi-objective performance improvements of general finite single-server queueing networks

Optimizing the performance of general finite single-server acyclic queueing networks is a challenging problem and has been the subject of many studies. The version of the optimization problem treated here considers the minimization of the buffer areas and the service rates simultaneously with the maximization of the throughput. These are conflicting objectives, and the most appropriate methodology appears to be a multi-objective methodology. In fact, algorithms have previously been proposed, and the aim here is to show that the use of a mixed methodology can occasionally improve solutions without a significant increase in the computational costs. This paper shows that improvements in throughput can be achieved through a solution of a type of stochastic knapsack problem, which consists of redistributing the buffer spaces between the lines while preserving the overall capacity using a simulated annealing algorithm; that is, one objective is improved (the throughput) without worsening the other (the overall allocated capacity). A set of computational experiments are presented to demonstrate the effectiveness of the proposed approach. Additionally, some of the insights presented here may help scientists and practitioners in finite single-server queueing network planning.

Asymptotic Throughput in Discrete‐Time Cyclic Networks with Queue‐Length‐Dependent Service Rates

For a discrete‐time closed cyclic network of single server queues whose service rates are non‐decreasing in the queue length, we compute the queue‐length distribution at each node in terms of throughputs of related networks. For the asymptotic analysis, we consider sequences of networks where the number of nodes grows to infinity, service rates are taken only from a fixed finite set of non‐decreasing sequences, the ratio of customers to nodes has a limit, and the proportion of nodes for each possible service‐rate sequence has a limit. Under these assumptions, the asymptotic throughput exists and is calculated explicitly. Furthermore, the asymptotic queue‐length distribution at any node can be obtained in terms of the asymptotic throughput. The asymptotic throughput, regarded as a function of the limiting customer‐to‐node ratio, is strictly increasing for ratios up to a threshold value (possibly infinite) and is constant thereafter. For ratios less than the threshold, the asymptotic queue‐length distribution at each node has finite moments of all orders. However, at or above the threshold, bottlenecks (nodes with asymptotically‐infinite mean queue length) do occur, and we completely characterize such nodes.

Single server tandem queues and queueing networks with non-correlated successive service times

To evaluate the local actual queueing delay in general single server queueing networks with non-correlated successive service times for the same customer, we start from a recent work using the tandem queue effect, when two successive local arrivals are not separated by “premature departures”. In that case, two assumptions were made: busy periods not broken up, and there are limited variations for successive service times. These assumptions are given up after having crossed two stages. The local arrivals become indistinguishable for the sojourn time inside a given busy period. It is then proved that the local sojourn time of this tandem queue effect may be considered as the sum of two components: the first (independent of the local interarrival time) corresponding to the case where upstream, successive service times are supposed to be identical to the local service time, and the second (negligible after having crossed 2 or 3 stages) depending on local interarrival times increasing because of broken up busy periods. The consequence is the possible occurrence of the agglutination phenomenon of indistinguishable customers in the buffers (when there are limited “premature departures”), due to a stronger impact of long service times upon the local actual queueing delay, which is not consistent with the traditional concept of local traffic source only generating distinguishable customers.

Single server queueing networks with varying service times and renewal input

Using recent results in tandem queues and queueing networks with renewal input, when successive service times of the same customer are varying (and when the busy periods are frequently not broken up in large networks), the local queueing delay of a single server queueing network is evaluated utilizing new concepts of virtual and actual delays (respectively). It appears that because of an important property, due to the underlying tandem queue effect, the usual queueing standards (related to long queues) cannot protect against significant overloads in the buffers due to some possible “agglutination phenomenon” (related to short queues). Usual network management methods and traffic simulation methods should be revised, and should monitor the partial traffic streams loads (and not only the server load).

Asymptotics of stochastic networks with subexponential service times

We analyse the tail behaviour of stationary response times in the class of open stochastic networks with renewal input admitting a representation as (\max,+)-linear systems. For a K-station tandem network of single server queues with infinite buffer capacity, which is one of the simplest models in this class, we first show that if the tail of the service time distribution of one server, say server i_0\in\{1,\ldots,K\}, is subexponential and heavier than those of the other servers, then the stationary distribution of the response time until the completion of service at server j\geq i_0 asymptotically behaves like the stationary response time distribution in an isolated single-server queue with server i_0. Similar asymptotics are given in the case when several service time distributions are subexponential and asymptotically tail-equivalent. This result is then extended to the asymptotics of general (\max,+)-linear systems associated with i.i.d. driving matrices having one (or more) dominant diagonal entry in the subexponential class. In the irreducible case, the asymptotics are surprisingly simple, in comparison with results of the same kind in the Cramer case: the asymptotics only involve the excess distribution of the dominant diagonal entry, the mean value of this entry, the intensity of the arrival process, and the Lyapunov exponent of the sequence of driving matrices. In the reducible case, asymptotics of the same kind, though somewhat more complex, are also obtained. As a direct application, we give the asymptotics of stationary response times in a class of stochastic Petri nets called event graphs. This is based on the assumption that the firing times are independent and that the tail of the firing times of one of the transitions is subexponential and heavier than those of the others. An extension of these results to nonrenewal input processes is discussed. Asymptotics of queue size processes are also considered.

Multiserver queueing networks and the tandem queue model

Using a tandem queue model we evaluate the local “endogenous” (= internal) queueing delay in single server and multiserver queueing networks. The new concept of the apparent overall upstream queueing delay(as perceived by the downstream network) allows us to analyze the distribution of this local queue by interpolating between the distributions of the tandem queue (generated by a concentration tree) and the isolated G/G/squeue. The interpolation coefficients depend on the proportion of “premature departures”, typically interfering in the upstream stage and leaving the considered path without being offered to the considered local queue. On the other hand, local “exogenous” arrivals (from outside the network) require the introduction of the “interference delay” concept. Finally, in the case of single server queueing networks, we stress the need to extend the capacities of the buffers, by considering the “worst case” scenario and by using an “equivalent tandem queue” model.

Congestion resulting from increased capacity in single-server queueing networks

In many networks for traffic and telecommunications, minimizing delays from entry to exit is a major concern of users. In user-optimal routing, each user chooses a path to minimize delay from entry to exit, given the existing paths chosen by all other users. Under user-optimal routing, at equilibrium all users experience the same delay. Many networks, especially data networks, are commonly modeled as networks of single-server queues. We report examples of single-server queueing networks with user-optimal routing in which adding servers or increasing the capacity of existing servers worsens the delay experienced by all users.

The theory of networks of single server queues and the tandem queue model

We consider the stochastic behavior of networks of single server queues when successive service times of a given customer are highly correlated. The study is conducted in two particular cases: 1) networks in heavy traffic, and 2) networks in which all successive service times have the same value (for a given customer), in order to avoid the possibility of breaking up the busy periods. We then show how the local queueing delay (for an arbitrary customer) can be derived through an equivalent tandem queue on the condition that one other local queueing delay is added: the jitter delay due to the independence of partial traffic streams.We consider a practical application of the results by investigating the influence of long packets on the queueing delay of short packets in modern packet switched telecommunication networks. We compare these results with the results given by traffic simulation methods to conclude that there is good agreement between results of calculation and of traffic simulation.

Scheduling in a multi-class series of queues with deterministic service times

We consider a problem of scheduling in a multi-class network of single-server queues in series, in which service times at the nodes are constant and equal. Such a model has potential application to automated manufacturing systems or packet-switched communication networks, where a message is divided into packets (or cells) of fixed lengths. The network is a series-type assembly or transfer line, with the exception that there is an additional class of jobs that requires processing only at the first node (class 0). There is a holding cost per unit time that is proportional to the total number of customers in the system. The objective is to minimize the (expected) total discounted holding cost over a finite or an infinite horizon. We show that an optimal policy gives priority to class-0 jobs at node 1 when at least one of a set ofm−1 inequalities on partial sums of the components of the state vector is satisfied. We solve the problem by two methods. The first involves formulating the problem as a (discrete-time) Markov decision process and using induction on the horizon length. The second is a sample-path approach using an interchange argument to establish optimality.

Optimal greedy policies for stochastic control models

We introduce the notion of a greedy policy for general stochastic control models. Sufficient conditions for the optimality of the greedy policy for finite and infinite horizon are given. Moreover, we derive error bounds if the greedy policy is not optimal. The main results are illustrated by Bayesian information models, discounted Bayesian search problems, stochastic scheduling problems, single-server queueing networks and deterministic dynamic programs.

Optimal intensity allocation of single-server queueing networks

Consider an optimal intensity allocation problem of a non-markovian queueing network with a single server. Both the arrival and service processes are modelled by point processes with stochastic intensities. All the stations share a single service facility. After finishing service at one station, a customer switches to another station or leaves the system according to specific probabilities. A linear holding cost is incurred at each station. The objective is to control the arrival intensity at each station and schedule the service intensity at all stations so that the expected discounted cost over an infinite horizon is minimized. The main result is that in the class of non-idling control policies, the optimal intensity is of a bang-bang type and the optimal scheduling rule turns out to be the Klimov's rule; whereas in the larger class of control policies that permits enforced idling, the optimal control rule is a modified static policy. The approach combines a point process calculus with dynamic programming

Fluid limits and diffusion approximations for networks of multi-server queues in heavy traffic

A Functional Strong-Law-of-Large-Numbers (known as fluid limit or fluid approximation) and a Functional Central Limit Theorem (known as diffusion approximation) are proved for both open and closed networks of multi-server queues in heavy traffic. The fluid limit is a reflected piecewise linear deterministic process, and the diffusion limit is a reflected Brownian motion; both limiting processes are on the nonnegative orthant for open networks and on the nonnegative unit simplex for closed networks. The results generalize the existing ones for networks of single-server queues.

Optimal control of single-server queueing networks

A general single-server queueing network model is considered. It is well-known that an optimal policy is determined by the largest-index policy. There is an index for each given queue and one allocates the server to a queue with largest current index. Using discounted dynamic programming we give a new and short proof of this result and derive some characterizations and bounds of the indices. Moreover, it is shown that an approximate largest-index policy yields an approximately optimal policy. These results lead to efficient methods for computing the indices. In particular, we present a general largest-remaining-index method.

The Evaluation of Normalizing Constants in Closed Queueing Networks

Recently P. G. Harrison derived a closed form solution for the normalizing constant of a closed network of single server queues. In the present paper, we describe an alternative method of obtaining Harrison's result. The new method can usefully be applied to other closed networks of queues, particularly those in which irregular constraints bound the state space. Examples of the method are provided, and several new results are obtained.

The influence of service-time variability in a closed network of queues

This paper describes the effect of service-time variability on the standard performance measures of a closed network of single-server queues with the first-come first-served discipline and one job class. Several service-time variability principles are proposed to serve as rough practical guidelines. The most interesting one states that the mean queue length at a bottleneck queue typically decreases when the variability of the service time at that queue is increased. The principles are supported here by numerical examples and theorems in special cases. The principles are also applied to test approximation procedures.

Performance of the Queueing Network Analyzer

This paper describes the performance of the Queueing Network Analyzer (QNA), a software package developed at Bell Laboratories to calculate approximate congestion measures for networks of queues. QNA is compared with simulations and other approximations of several open networks of single-server queues. This paper illustrates how to apply QNA and indicates the quality that can be expected from the approximations. The examples here demonstrate the importance of the variability parameters used in QNA to describe non-Poisson arrival processes and nonexponential service-time distributions. For these examples, QNA performs much better than the standard Markovian algorithm, which does not use variability parameters. The accuracy of the QNA results (e.g., the expected delays) in these examples is satisfactory for engineering purposes.

On Homogeneity and On Line=Off-Line Behavior in M/G/1 Queueing Systems

Operational analysis replaces certain classical queueing theory assumptions with the conditions of homogeneous service and on-line= off-line behavior. In this paper we explore the relationship between the operational and classical concepts for the sample paths of an M/G/1 queueing system. The primary results are that the sample paths can have these operational properties with nonzero probability if and only if the service time is exponential. We also state dual results for interarrival times in G/M/l. Additionally, we show that open, feedforward networks of single server queues can have product form solutions valid across a range of system arrival rates if and only if all of the service times are exponential. Finally, we consider the relationship between the operational quantities S(n) and the mean service time in M/G/1. This relationship is shown to depend on the form of the service time distribution. It follows that using operational analysis to predict the performance of an M/G/1 queueing system will be most successful when the service time is exponential. Simulation evidence is presented which supports this claim.

Stochastic Models for Multistage Cell Classification Systems

Probabilistic models for multistage cell classification systems are described. A simple finite Markov chain models classification events which occur as a cell passes through the system. The state space consists of various identities assigned to the cell, including true celi type and identities assigned by classifiers. Effects of throughput rate, data buffer capacity, and classifier processing rate on system performance are predicted by another model composed of a network of single server queues. Markov and queue models are interrelated in that classification events at one processor (modeled by the Markov chain) govern arrival rates of other processors. In turn, the queue model predicts the probability that a cell wili be missed due to fmite data buffer capacity. The miss event is modeled by the Markov chain as a possible classification outcome. Application of the models is illustrated for a multistage gynecologic flow prescreening system with slit-scan processing in the first stage and two dimensional image processing in the second. Results predict system sensitivity as a function of first stage false alann rate and abnormal cell occurrence rate.