Importance sampling for Jackson networks with customer impatience until the end of service

Abstract

Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. The standard approach, which simulates the system using an a priori fixed change of measure, has been shown to fail in even the simplest network settings. Estimating probabilities associated with rare events has been a topic of great importance in queuing theory, and in applied probability at large. In this paper, we estimate the probability of two rare events known as total population overflow and individual buffer overflow in an open Jackson network in which the customers should receive the needed service in a definite deadline. we use parallel computing in implementing the estimator. Moreover, we consider the effect of various network parameters on aforementioned overflow probabilities, and we have also shown that how these parameters affect the probability of missing the deadline.

Introduction

Rare event simulation has drawn the attention of scholars since the development of Monte Carlo techniques on computers at Los Alamos National Laboratory (Rubino and Tuffin, 2009). It involves estimating extremely small but significant probabilities. In spite of the extent of the research on the topic in past decades, there are still areas waiting to be studied due to new applications (Kroese et al., 2011). The estimation of rare event probabilities has been attractive in queueing theory. In the wake of their generic structure, queues proved to be of great importance in applied probability, as various applications may be found in inventory, military, communication networks, call centers, etc. The main motivation for studying these rare events lies in the fact that the events under consideration relate to situations whose happening can be very costly for queue network to face. To cite some usual examples, we may refer to the claims made by an insurance company which exceeds its capital, customers' complaint over long delay in call centers, or performance breakdown in a communication network. The common aspect in these examples is that the rare event under consideration can be rephrased as the event that the workload in a chosen queuing system exceeds a specific high threshold.

Importance sampling has considered the expression as the most common approach in rare event simulation. It is believed that importance sampling changes the sampling distribution of the system under study in order to sample more frequently the events that are more “important” for the simulation. However, employing a new distribution will lead to a biased estimator if no correction is considered. Hence, by multiplication with the so-called likelihood ratio (Rubino and Tuffin, 2009) the simulation output is converted to the original measure.

In rare event simulation in general and importance sampling, in particular, the problem of estimating the probabilities of total population overflow and individual buffer overflow of Markovian queues has been an important benchmark problem (de Boer and Scheinhardt, 2010). The existing research papers on the application of importance sampling in queueing ordinarily depend on a state-independent change of measure. Another word, for any system state, the probabilistic law is changed in the same way (de Boer, 2006). It was believed that the main problem of state-independent importance sampling schemes was that the transition rates changed in a uniform manner, irrespective of the fact that one of the queues was empty or not (Glasserman and Kou, 1995). Thereby, there is no assurance that the likelihood ratio is bounded on the event of interest. Therefore, the performance of importance sampling scheme proposed in Parekh and Walrand (1989) is far from ideal for some parameter values. It is worth mentioning that de Boer et al. (2000) and Kroese and Nicola (2002) suggested some solutions to solve this problem where, state-dependent importance sampling schemes were proposed, 1i.e., importance sampling distributions that are not uniform over the state space. Of great importance is the study conducted by Dupuis et al. (2007a). This study, which was the first of its kind, proved the asymptotic efficiency for a state-dependent importance scheme for estimating overflow probabilities in Jackson networks. In addition, a state-dependent importance sampling scheme for estimating the probability of overflow in the downstream queue of a Jacksonian two-node tandem queue was developed in Miretskiy et al. (2010). The robustness of a discrete-time Markov chain was checked in L'ecuyer et al. (2010) where state-dependent importance sampling was employed to estimate first-passage time probabilities.

In all the previous literature, models of queueing networks have been considered in which, each customer enters the network and waits to receive service without any time limitation. Once it receives the service from the different nodes of the network, it leaves it. According to such method, a number of schemes have been raised for simulating rare events in the various network models. In these models, customers do not rush to receive service since receiving the service is important. This type of modeling is used in different applications such as some packet-switched telecommunication networks where no time period is guaranteed for packet transfer, and the transfer time depends on the traffic load as well as the network parameters. However, there are other models in which customers wait for service, but for a limited time. In these models, as in the previous papers (Mahdipour and Rahmani, 2009, Mahdipour et al., in preparation, Mahdipour et al., 2009) we have analyzed a simple case of it, the service time for a customer would be very important. In one node, the sojourn time of a customer should not exceed a definite deadline, and if it does, the customer would not wait to receive the service from that node and therefore leaves the network. This type of situation happens in the processing or merchandizing of perishable goods (Barrer, 1957a). It is true for many types of military engagements. For example, an attacking airplane engaged by antiaircraft is available for ‘service’ – within the range – for only a limited time.

The contribution of our work consists of three main parts. Firstly, we estimate the probability of two rare events known as total population overflow and individual buffer overflow in an open Jackson network in which the customers should receive the needed service in a definite deadline. Secondly, using parallel computing we present a rapid estimator which has the capability of being used in real time applications. Thirdly, and more importantly, we consider the effect of various network parameters on aforementioned overflow probabilities, and we have also shown that how the various network parameters affect the probability of missing the deadline.

We have not encountered a modeling that provides estimation of the probability of occurrence of rare events in networks where the sojourn time of customers is limited.

The paper is organized as follows. Section 2 presents a brief review of related work. In Section 3 an analysis of the probability of missing the deadline in a K-node Jackson network and its dynamics are derived. Section 4 introduces the rare event probabilities in a K-node Jackson network. The dynamic importance sampling schemes are defined in Section 5 and an appropriate state-dependent importance sampling scheme is constructed in Section 6. Numerical results are presented in Section 7. The paper concludes with Section 8.