Stability of Queueing Networks

This paper considers an M/M/1 queueing system with dynamically controlled arrival and service rates. At each arrival or service completion epoch, a decision maker chooses a pair of arrival and service rates from a finite set, and the system operates under these rates until the next arrival or service completion. There is a switching cost for changing the rates, and there is a cost per unit time of holding customers and using the arrival and service rates. The results describe natural conditions on the costs under which an optimal policy for either the discounted-cost or average-cost criterion is a hysteretic policy. Such a policy increases the service rate and decreases the arrival rate as the queue length increases. Furthermore, the control exhibits a stickiness or resistance to change, called hysteresis, due to the switching costs.

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all work conserving policies.However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations we prove that if the fluid model is not weakly stable under the class of all work conserving policies, then any corresponding queueing network is not rate stable under the class of all work conserving policies. We establish the result by building a particular work conserving scheduling policy which makes any corresponding stochastic process transient. An important corollary of our result is that the condition of the form ρ* < 1, which was proven in [7) to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for any corresponding queueing network. Here ρ* is a certain computable parameter of the network involving virtual station and push start conditions.

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all nonidling policies.\n¶\nHowever, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations, we prove that if the fluid model is not weakly stable under the class of all nonidling policies, then a corresponding queueing network is not rate stable under the class of all nonidling policies. We establish the result by building a particular nonidling scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition ρ<sup>*</sup>≤1, which was proven in [Oper. Res. 48 (2000) 721–744] to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here ρ<sup>*</sup> is a certain computable parameter of the network involving virtual station and push start conditions.

In this paper, two multi-objective models to optimally control the lead time in multi-server multi-stage assembly system by considering server allocation problem (SAP) and also service rate control problem (SCP) are presented, where new product orders, including all their operations, are entered to the system according to a Poisson process, only one type of products also is produced by the production system, and multi-servers can be settled in each service station. Each operation of any work is operated at a devoted service station with only one of the servers located at a node of the network based on a first-come-first-serve discipline, while the processing times are independent random variables with exponential distributions. Furthermore, it is also assumed that the transport times between each pair of service stations are independent random variable with generalized Erlang distributions. Such system can be modeled as a queueing network, where the system is in the steady state and the lead time is controllable. For modeling of multi-server multi-stage assembly system, initially the network of queues is transformed into an appropriate stochastic network with exponentially distributed arc lengths. A differential equations system is organized to solve and obtain approximate lead time distribution for any particular wok by applying a proper finite-state continuous-time Markov model. Then, two multi-objective models including four conflicted objectives are presented to optimally control the servers allocated to the service stations in SAP and also service rate of service stations in SCP. For solving a discrete-time approximation of the primary multi-objective problems, the goal attainment technique is employed. In this research, reactive controlling in a multi-server multi-stage assembly system also is discussed.

To analyze networks of queues, it is important to be able to analyze departure processes from single queues. For the M/M/s and M/G/∞ models, the stationary departure process is simple (Poisson), but in general the stationary departure process is quite complicated. As a basis for approximations, this paper shows that the stationary departure process is approximately Poisson when there are many busy slow servers in a large class of stationary G/GI/s congestion models having s servers, infinite waiting room, the first-come first-served discipline, and mutually independent and identically distributed service times that are independent of a stationary arrival process. Limit theorems are proved for the departure process in a G/GI/s system in which the number of servers and the offered load (arrival rate divided by the service rate) both increase. The asymptotic behavior of the departure process depends on the way the arrival rate changes. If the arrival rate is held fixed, so that the offered load increases by slowing down the service rate, then the departure process converges to a Poisson process. For this result, the service-time distribution is assumed to be phase-type. Other limiting behavior occurs if the arrival rate approaches zero or infinity. Convergence is established in each case by applying previous heavy-traffic limit theorems.

We consider a model for a single link in a circuit switched network. The link has C circuits, and the input consists of offered calls of two types, that we call primary and secondary traffic. Of the C links R are reserved for primary traffic. We assume that both traffic types arrive as Poisson arrival streams. Assuming that C is large and R=O(1), the arrival rate of secondary traffic is O(C) while that of primary traffic is smaller, of the order . The holding times of the secondary calls are assumed exponentially distributed with unit mean. Those of the primary calls are exponentially distributed with a large mean, that is . The loads for both traffic types are thus comparable (O(C)) and we assume that the system is “critically loaded,” i.e., the system's capacity is approximately equal to the total load. We analyze asymptotically the steady state probability that n2 (resp. n1) circuits are occupied by primary (resp. secondary) calls. In particular, we obtain two‐term asymptotic approximations to the blocking probabilities for both traffic types. This work complements part I, where we assumed that the secondary traffic had an arrival rate that was and a service rate that was . Thus in part I the R trunks were reserved for the “fast traffic,” whose arrival and service rates were O(C) and O(1).

This study presents a comprehensive analytical investigation of a four-server queueing system, modeled as a priority feedback queue network operating under stochastic conditions. The system architecture comprises a primary server that incorporates a priority mechanism and is connected to three parallel servers. Customers, divided into low- and high-priority groups, arrive at the primary server according to a Poisson process. Service times at all servers are assumed to follow exponential distributions. After completion of service at any server, a customer will either exit the system or re-enter for further service with a different feedback probability, which reflects a realistic service satisfaction mechanism. This analysis evaluates explicit expressions for time-independent state probabilities and derives steady-state performance metrics of the network, including server utilization, average queue lengths, waiting times, and throughput by application of generating function techniques and the calculus-based resolution of the associated differential-difference equations. Extensive numerical simulations are carried out to validate the theoretical results and to delve into the impact of different parameters on various performance indicators, which gives information to system administration in order to reduce congestion. This study yields actionable insights into the optimal allocation of resources, aiming to minimize delays and maximize throughput under varying stochastic conditions. Furthermore, the behavioral analysis of the model underscores the notable effects of priority and feedback mechanisms on the system achievement measures and provides a foundation for optimization strategies in multi-class service environments. The findings have significant implications for the design and optimization of complex queueing networks in telecommunications, manufacturing systems, and other service-oriented industries where feedback and priority-based processing are prevalent.

Formulating the solution as an optimisation problem has proven to be effective in developing solutions to many real world problems. We generally obtain the best possible solution using these methods. In this work, the traffic scheduling problem has been formulated as a waiting time minimisation problem, and appropriate cost functions have been developed, in pursuit of finding the optimal solution. A first-in, first-out queuing model is used, with the vehicles arriving in a Poisson process, and the service time being exponentially distributed. The key feature of this model is that it adapts to varying service and arrival rates of the lanes. These rates are forecast using a neural network model, and appear in the objective function. It was observed that the use of the neural network greatly improved the robustness of the model. Although the model has been developed for a four lane, two way architecture, it can be generalised to any architecture. Results have been analysed by comparing the proposed method to a proportional time distribution. It is shown that the proposed model performs relatively well, when there is rapid variation in the arrival and service rates.

A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

Queueing Network Models (QNMs) with Finite Capacity provide powerful and realistic tools for the performance evaluation and prediction of discrete flow systems such as computer systems, communication networks and flexible manufacturing systems. Over recent years, there has been a great deal of progress towards the analysis and application of QNMs with finite capacity, and high quality research work has appeared in diverse scientific journals of learning and conference proceedings in the fields of Operations Research, Computer Science, Telecommunication Networks, Management and Industrial Engineering. However, there are still many important and interesting finite capacity queues and QNMs to be resolved, such as those involving multiple-job classes, bounds and theoretical properties, exact analysis, numerical solutions and approximate methods, as well as application studies to computer and distributed systems, high-speed networks and production systems.This volume of Annals of Operations Research on Queueing Networks with Finite Capacity presents 17 extended research papers, both from industry and academia, reflecting current original contributions in the theory and applications of finite capacity queues and networks worldwide. Earlier short versions of these papers were amongst those presented at QNets '95, The Third International Workshop on Queueing Networks with Finite Capacity, 6th-7th July, 1995, Ilkley, West Yorkshire, UK. The papers are classified into four parts covering exact analysis, approximate methods, error bounds and performance modelling studies.

This paper proposes a procedure for constructing the membership functions of performance measures in finite-capacity queuing systems where both arrival and service rates are represented as fuzzy numbers. By applying the (\alpha)-cut method, a fuzzy queue with finite capacity is transformed into a family of conventional crisp queues, allowing for more precise modeling of system characteristics. The study focuses on a queuing model with an unreliable server, where key parameters such as service and breakdown rates are fuzzy values. The developed parametric nonlinear programming approach facilitates the derivation of new constraints, providing a robust framework for analyzing queuing behaviors under uncertainty. The findings demonstrate that the proposed fuzzy mathematical model yields more realistic outcomes than traditional crisp models, thereby enhancing the applicability of queuing theory in practical scenarios.

Heavy traffic limit theorems are established for a class of single server queueing models including those with heavy-tailed or long-range dependent arrivals and time-varying service rates. The models are motivated by wireless queueing systems for which there is an increasing evidence of the presence of heavy-tailed or long-range dependent arrivals, and where the service rates vary with the changes in the wireless medium. The main focus of the paper is to obtain the different possible limit processes that can arise depending on the relationship between scalings for both the arrival and departure processes. The limit processes obtained here are driven by either Brownian motion (when the contribution from the departure process dominates the limit) or the limits of properly scaled arrivals (when the contribution from the arrival process dominates the limit), typical examples being stable Lévy motion or fractional Brownian motion. In particular, for the case where arrival process is given by the infinite source Poisson process, this relationship, which determines the type of the limiting queue-length process, is a simple condition involving the heavy tail exponent, arrival rate and channel variation parameter in the wireless medium model. To establish these limit results, two approaches are studied. In one approach, when the limit is driven by Brownian motion, the perturbed test function method is extended to incorporate reflection. In contrast, the second approach allows for non-Markovian and/or non-Gaussian driving processes in the limit. Both approaches involve averaging in the drift term arising from random service rates at the departures. In the second approach, this averaging is carried out directly and pathwise, thus sidestepping the assumption of driving Brownian motion used in the perturbed test function method.

This paper concentrates on single server bulk arrival queue system with batch size dependent service and working vacation. The server provides service in two service modes depending upon the queue length. The server provides single service if the queue length is at least ‘a’. On the other hand the server provides fixed batch service if the queue length is at least ‘k’ (k > a). Batch service is provided with some fixed batch size ‘k’. After completion of service if the queue length is less than ‘a’ then the server leaves for working vacation. During working vacation customers are served with lower service rate than the regular service rate. Service during working vacation also contains two service modes. For the proposed model probability generating function of the queue length at an arbitrary time will be obtained by using supplementary variable technique. Some performance measures will also be presented with suitable numerical illustrations.

We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the nodes is exponentially distributed. After service completion at a node, the serviced customer can transit to another node or leave the network forever. The main features of the model are the mutual dependence of processes of customer arrivals and retrials and the impatience and non-persistence of customers. Dynamics of the network are described by a multidimensional Markov chain with infinite state space, state inhomogeneous behavior and special structure of the infinitesimal generator. The explicit form of the generator is derived. An effective algorithm for computing the stationary distribution of this chain is recommended. The expressions for computation of the key performance measures of the network are given. Numerical results illustrating the importance of the account of the mentioned features of the model are presented. The model can be useful for capacity planning, performance evaluation and optimization of various wireless telecommunication networks, transportation and manufacturing systems.