A note on networks of infinite-server queues

Call-blocking probabilities are among the key performance measures in mobile communications networks. For their analysis, mobile networks can be modelled as networks of Erlang loss queues with common capacity restrictions dictated by the allocation of frequencies to the cells of the network. However, due to the time-varying load offered to the cells of such networks, blocking probabilities usually cannot be obtained in closed form. The relation between networks of Erlang loss queues and networks of infinite server queues, for which the time-dependent occupancy distribution is multidimensional Poisson, suggests to use that distribution as approximate distribution for the network of Erlang loss queues. This paper extends this so-called Modified Offered Load (MOL) approximation to networks of Erlang loss queues, and also allows subscribers that find their call blocked to redial to continue their call. For GSM networks operating under Fixed Channel Allocation, it is shown that blocking probabilities are increasing in the redial rates so that the MOL approximation that is most accurate for maximal redial rates turns out to be fairly accurate for the resulting upper bound for blocking probabilities. The accuracy is explicitly evaluated in an application of the results towards blocking probabilities in a hot spot travelling along a road through a GSM network.

Networks of infinite-server queues with multiplicative transitions

We consider a network of infinite-server queues where the input process is a Cox process of the following form: The arrival rate is a vector-valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. We first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above-mentioned Cox process. We also obtain transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. Finally, we extend to the setting of a network of infinite-server queues.

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.

Queueing networks have been widely used to evaluation performance of mainframe computer systems. In contrast, few results have been reported for modern open systems, so it was not clear whether queueing networks are useful for modern systems or not. We think this situation has partly been due to lack of handy evaluation tools. This paper presents tow tools that we developed to evaluate open system performance. On is a measuring tool that is capable of accurately obtaining the service times of system resources requested by an application transaction. The other is an estimating tool which calculates various performance measures based on queueing network models. This paper also describes a case study in which the performance of a medium‐sized UNIX client–server system (up to 24 clients) is estimated using the tools and these estimates are then compared with experimental results. The estimates closely agree with the measured results and are accurate enough for practical applications. Based on this, we conclude that queueing network models are also useful for modern systems.

We consider networks of queues in discrete time, where the steady state distribution can be computed explicitly in closed form (product form networks): (i) Closed cycles and open tandems of single server FCFS Bernoulli nodes with state dependent service probabilities, where customers flow linearly, (ii) networks of doubly stochastic and geometrical queues (which are discrete time analogues of Kelly’s symmetric, resp. general, servers), where customers of different types move through the network governed by a general routing mechanism and request for service according to general, resp. geometrical, distributions, (iii) networks with batch movements of customers and batch service, where the service and routing mechanism is defined via an abstract transition scheme.

We study G-networks with positive and negative customers and signals. We consider two types of signals: they can make a subnetwork of queues operational or down. As signals are sent by queues after a customer service completion, one can model the availability of a sub-network of queues controlled by another network of queues. We prove that under classical assumptions for G-networks and assumptions on the rerouting probabilities when a subnetwork is not operational, the steady-state distribution, if it exists, has a product form steady state distribution. Some examples are given.

This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

Este artigo ilustra a aplicação de modelos de redes de filas abertas para auxiliar no processo de tomada de decisões de projeto (ou reprojeto) e planejamento de sistemas discretos de manufatura. Utilizam-se métodos aproximados de decomposição paramétrica, para avaliar as medidas de desempenho da rede. Por meio de um estudo de caso de uma rede job-shop de uma fábrica metal-mecânica, mostra-se que estes métodos são adequados e efetivos para estimar, por exemplo, os estoques em processo (WIP) e os leadtimes de produção dos produtos. Mostra-se ainda que modelos de programação matemática de alocação de capacidade nas estações da rede, baseados nos métodos de decomposição, também são efetivos para otimizar o desempenho da rede em diferentes configurações. Curvas de trade-off entre investimento em capacidade e WIP ou leadtime são geradas e são úteis para auxiliar um gerente a decidir quanto alocar de capacidade na rede para atingir metas de desempenho. Estas curvas também são úteis para analisar o efeito em termos de capacidade se a variabilidade das chegadas externas de produtos, no mix de produtos ou na taxa de produção da rede muda.

Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.

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.

The present paper is devoted to the research of the mathematical model of an insurance company in the form of the queueing system with an infinite number of servers. The arrival process of risks is regarded as a modulated Poisson arrival process. Applying the asymptotic analysis method under the condition of a high-rate arrivals, the characteristic function of the probability distribution for the two-dimensional process of the number of risks and the number of claims for insurance payments is obtained. It is shown that this probability distribution can be approximated by Gaussian distribution. These results can be applied to the estimation of functioning of the various economic systems.

This paper investigates a queuing system with infinite number of servers where the arrival process is given by a Markov Arrival Process (MAP) and the service time follows a Phase-type (PH) distribution. They were chosen since they are simple enough to describe the model by exact methods. Moreover, highly correlated arrival processes and heavy-tailed service time distributions can be approximated by these tools on a wide range of time-scales. The transient behaviour of the system is analysed and the time-dependent moments of the queue length is computed explicitly by solving a set of differential equations. The results can be applied to models where performance of parallel processing is important. The applicability of the model is illustrated by dimensioning a WEB-based content provider.

For a network ofG/∞ service facilities, the transient joint distribution of the facility populations is shown by new simple methods to have a simple Poisson product form with simple explicit formulas for the means. In the network it is assumed that: (a) each facility has an infinite number of servers; (b) the service time distributions are general; (c) external traffic is non-homogeneous in time; (d) arrivals have random or deterministic routes through the network possibly returning to the same facility more than once; (e) arrivals use the facilities on their route sequentially or in parallel (as in the case of a circuitswitched telecommunication network). The results have relevance to communication networks and manufacturing systems.

Queueing network [formula omitted] with high-rate arrivals