Closed-form formulae for moment, tail probability, and blocking probability of waiting time in a buffer-sharing deterministic system

Abstract

Obtaining analytic expressions for characteristics in probabilistic systems with finite buffer capacities such as (higher) moments and tail probabilities of stationary waiting times, and blocking probabilities is by no means trivial. This is also true even for a system with deterministic processing times. By using the max–plus algebraic approach in this study, we introduce closed-form formulae for characteristics of stationary waiting time in a complete buffer-sharing m-node tandem system with constant processing times. Numerical examples are also provided.

Introduction

For design and maintenance purposes, it is sometimes necessary to investigate the queueing mechanisms embedded in various computer systems, telecommunication networks, and automated manufacturing systems. Although finite-capacity queueing systems have been widely studied, research in this area has rendered few explicit results. Due to the difficulties posed by finite capacities, analytic solutions are difficult to obtain; most studies have been limited in number of nodes, distributions of arrivals, service times, and so on. There are, however, a few analytic solutions to finite-capacity queues for special cases such as $M∕M∕1∕K$, $M∕G∕c∕c$, $M∕G∕1∕1$, $M∕G∕1∕2$, and so on. Tijms [14] presented recursive formulae for stationary distributions and blocking probabilities in $M∕G∕c∕K$ queues (also see Takagi [13]). Brun and Garcia [6] derived analytical (transform-free) solutions for steady-state distribution in $M∕D∕1∕K$ queues by using a generating function. The standard queueing theory is not yet applicable, however, to general queues such as multi-node, multi-server, and generally structured queues. To achieve analytic solutions for multi-node systems, most studies have used decomposition methods by decomposing the network into subnetworks and treating each subnetwork independently with adjusted parameters such as input rates and service rates (see, e.g., Jun and Perros [7]; Shi [12]; and the references therein).

Unlike the infinite buffer case, the distribution of waiting time in tandem queues with a finite buffer is not simply given as a product form due to the blocking phenomena between nodes. Therefore, various approximation methods using decomposition and simulation have been proposed. In this study, however, we develop an exact solution procedure based on max–plus algebra. The max–plus linear system uses only two operators, “max” and “plus”, to represent its performance characteristics. It is well known that the max–pluslinear system (MPL) includes various probabilistic systems commonly found in telecommunication and computer networks. Ayhan and Seo [1], Baccelli et al. [3] provided some preliminaries on max–plus algebraic representation of waiting times in MPLs.

Conceptually, a buffer sharing policy can be applied to various systems without system configuration limitations. However, we here focus on a tandem system consisting of $m$ nodes having constant processing times and having a Poisson arrival process with rate $\lambda$ in order to obtain analytical solutions for waiting time perspective.

Two typical blocking policies adopted in many researches are communication blocking (blocking before service) and production blocking (blocking after service). Under a production blocking policy the common buffer is occupied in advance by a blocked job waiting at node 0 (a dummy node). However, under a communication blocking policy the common buffer is occupied only when a blocked job becomes unblocked and is joining in node 1 (the first node in our actual system). Communication blocking is more suitable for representing the blocking phenomena prevalent in general queueing systems and has simpler expressions in max–plus algebraic notation than production blocking. Thus it is assumed that pulling a job between nodes 0 and 1 in the system follows a communication blocking policy. Under a complete buffer sharing policy, we introduce explicit expressions for higher moments and tail probability of stationary waiting times in an $m$-node tandem system with constant processing times, and also obtain a closed-form formula for blocking probability.