Nam K. Kim

A Two-Moment Approximation for the GI/G/c Queue with Finite Capacity

In this paper, we consider the steady-state queue length of the multiserver finite-capacity GI/G/c/c+r queue. As a result, we first obtain an exact transform-free expression for the steady-state queue-length distribution. Making use of this result, we then present a simple two-moment approximation for the queue-length distribution. From this, approximations for some important performance measures, such as the loss probability, the mean queue length, and the mean waiting time, are also obtained. In addition, we propose an approximation for the minimal buffer size that keeps the loss probability below an acceptable level. Extensive numerical experiments show that our approximation is extremely simple yet fairly good in its performance.

An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions

For a broad class of discrete- and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some multiserver queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the probability generating functions (PGFs) of the stationary numbers in queue and in system, as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with batch Markovian arrival process (BMAP) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.

Equivalences of Batch-Service Queues and Multi-Server Queues and Their Complete Simple Solutions in Terms of Roots

Abstract In this article, we first present a unified discussion of several equivalence relationships among (as well as between) batch-service queues and multi-server queues, in terms of the stationary queue-length and waiting-time distributions. Then, we present a complete and simple solution for the queue-length and waiting-time distributions of the discrete-time multi-server deterministic-service Geo/D/b queue, in terms of roots of the so-called characteristic equation. This solution also represents the solutions for the other equivalent queues, as a result of the equivalence relationships. To aid in the applications of these results, sample numerical results are presented at the end.

The use of the distributional Little's law in the computational analysis of discrete-time GI/G/1 and GI/D/c queues

In this paper, we first establish a discrete-time version of what is called the distributional Littles law, a relation between the stationary distributions of the number of customers in a system (or queue length) and the number of slots a customer spends in that system (or waiting time). Based on this relation, we then present a simple computational procedure to obtain the queue-length distribution of the discrete-time GI/G/1 queue from its waiting-time distribution, which is readily available by various existing methods. Using the same procedure, we also obtain the queue-length distribution of the discrete-time multi-server GI/D/c queue in a unified manner. Sample numerical examples are also given.

Remarks on the remaining service time upon reaching a target level in the M/G/1 queue

In this note we establish connections between new and previous results on the remaining service time upon reaching a target level in the M/G/1 queue.

On the Distribution of the Number of Customers in the D-BMAP/Ga,b/1/M Queue – A Simple Approach to a Complex Problem

Abstract In this paper, we consider a single-server finite-capacity queue with general bulk service rule where customers arrive according to a discrete-time batch Markovian arrival process (D-BMAP). The model is denoted by D-BMAP/Ga,b/1/M which includes a wide class of queueing models as special cases. We give a relation between the steady-state probabilities of the number of customers in the queue at departure- and arbitrary-epochs using the concept of the mean sojourn time in the phase of the underlying Markov chain of D-BMAP before the next arrival. We use the embedded Markov chain technique to obtain the departure-epoch probability of the number of customers in the queue. The pre-arrival probability of the number of customers in the queue is also obtained. Finally, a complete solution to the distribution of the number of customers in the D-BMAP/Ga,b/1/M queue, some computational results, and performance measures such as loss probability and mean queue length are presented.

On the Queue Length Distribution for the GI/G/1/K/VM Queue

Abstract We present a transform-free distribution of the steady-state queue length for the GI/G/1/K queueing system with multiple vacations under exhaustive FIFO service discipline. The method we use is a modified supplementary variable technique and the result we obtain is expressed in terms of conditional expectations of the remaining service time, the remaining interarrival time, and the remaining vacation, conditional on the queue length at the embedded points. The case K → ∞ is also considered.

Analysis of a Discrete-Time Finite-Capacity Single-Server Queue with Markovian-Arrival Process and Random-Bulk-Service: D-MAP/GY/1/M

In this article, we consider a single-server, finite-capacity queue with random bulk service rule where customers arrive according to a discrete-time Markovian arrival process (D-MAP). The model is denoted by D-MAP/G Y /1/M where server capacity (bulk size for service) is determined by a random variable Y at the starting point of services. A simple analysis of this model is given using the embedded Markov chain technique and the concept of the mean sojourn time of the phase of underlying Markov chain of D-MAP. A complete solution to the distribution of the number of customers in the D-MAP/G Y /1/M queue, some computational results, and performance measures such as the average number of customers in the queue and the loss probability are presented.