Closed-form formulae for moment, tail probability, and blocking probability of waiting time in a buffer-sharing deterministic system
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 , , , , and so on. Tijms [14] presented recursive formulae for stationary distributions and blocking probabilities in queues (also see Takagi [13]). Brun and Garcia [6] derived analytical (transform-free) solutions for steady-state distribution in 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 nodes having constant processing times and having a Poisson arrival process with rate 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 -node tandem system with constant processing times, and also obtain a closed-form formula for blocking probability.
Section snippets
Explicit expression for moments of waiting time
First we introduce brief preliminaries on max–plusalgebraic approach. Baccelli and Schmidt [5] introduced that the dynamics of max–plus linear systems with nodes can be described by the -dimensional vectorial recurrence equations with an initial condition of , where the refers to maximization and the refers to addition for scalars and max–plus algebra product for matrices, is a non-decreasing sequence of real-valued random numbers (e.g. the epochs of the
Blocking probability
Blocking probability for arrivals is a very attractive measure in the analysis of finite-capacity queues (see Fig. 1). Recently, Seo et al. [11] introduced an approximation computing method for blocking probability in a two-node tandem queue in which each node has a finite buffer capacity and constant processing time. In order to calculate blocking probability, they used the following simple blocking probability formula for an queue with the mean arrival rate : where
Optimization problem for QoS
In this subsection we consider an optimization problem that determines the minimum required buffer capacity for a given service level. Response (or sojourn) time is an important measure for service quality. From (9), (9), and (10), we can consider QoS only at upstream nodes of the bottleneck node. Let and be a pre-specified bound on the stationary waiting time and a required service level, respectively. Since the departure process of node in each buffer-sharing system is not
Examples and simulations
Because the explicit expression for higher moments given in Corollary 3.1 of Ayhan and Seo [1] involves the polynomial terms (see (7)), it needs much more computational time than our new expression, Theorem 1, in which the polynomial terms are eliminated. In addition, the computational time in our new expression is not sensitive to system parameters such as the number of nodes, finite buffer capacity, and service times.
In this section we consider a buffer-sharing 5-node tandem system
Acknowledgment
This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03934690).
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