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A finite-buffer G I / M / 1 / N − type queueing model is considered. The explicit formula for the Laplace transform of the transient queue-size distribution, conditioned by the number of packets present in the system at the starting time, is derived. The shape of the formula allows for finding the stationary distribution by applying the key renewal theorem. Moreover, the convergence rate of the transient queue-size distribution to the stationary one is determined with the constant value given explicitly. Numerical example is attached as well.

We present a Markov model to analyze the queueing behavior of the nonstationary G ( t ) / G ( t ) / s ( t ) + G ( t ) queue. We assume an exhaustive service discipline (where servers complete their current service before leaving) and use acyclic phase-type distributions to approximate the general interarrival, service, and abandonment time distributions. The time-varying performance measures of interest are: (1) the expected number of customers in queue, (2) the variance of the number of customers in queue, (3) the expected number of abandonments, and (4) the virtual waiting time distribution of a customer arriving at an arbitrary moment in time. We refer to our model as G-RAND since it analyzes a general queue using the randomization method. A computational experiment shows that our model allows the accurate analysis of small- to medium-sized problem instances.

This paper analyzes the waiting time distribution in the M/PH/1 retrial queue. We give expressions for the Laplace-Stieltjes transform of the waiting time. We then provide a numerical algorithm for calculating the Laplace-Stieltjes transform of the waiting time. Numerical inversion of the Laplace-Stieltjes transforms is used to calculate the waiting time distribution. Numerical results are presented to illustrate our results.

In this paper, we consider correlated queues where the service time of a packet is strongly correlated with its interarrival time due to the finite transmission capacity of input and output links. We present an analytical method to derive the LST (Laplace Stieltjes Transform) of the (actual) waiting time distribution and the system time distribution. To investigate the impact of such correlation between service and inter-arrival times on the system performance, we consider a counterpart GI/G/1 queue where the service time and inter-arrival time distributions are the same as in our correlated system, but they are independent. Some numerical examples are provided to show that such correlation gives significant impact on the system performance.

Measurements of file sizes transported on the World-Wide-Web have led some researchers to propose describing them by probability distributions with infinite variance. The M/G/1 queue often arises as a performance model for components of the WWW, and the service times correspond to file sizes; the infinite variance of the file sizes becomes the variance of the service times. In this paper the effects of very large service-time variances on some performance measures for the M/G/1 queue are explored via numerical examples and analytic arguments. The first main conclusion is that it is the form of the service-time distribution over a wide finite range that controls the steady-state queueing performance, so distributions with very large finite variances can yield the same behavior as distributions with infinite variances. The second main conclusion is that very large service-time variances cause the rate of approach to steady-state performance to be so slow that steady-state performance measures are not likely to be of engineering interest. A third conclusion is that a common device of using the probability that the work in an infinite queue exceeds the level b to approximate the probability that a finite buffer of size b overflows may be very inaccurate. The approximation works better for the fat-tailed distributions studied than for the others. The most important engineering implication of these results is that when service times have a very large variance (such as file transfers on the WWW), performance criteria other than steady-state measures have to be used.

A Bernoulli schedule is a non-exhaustive service discipline for cyclic-service queuing systems (polling systems) which operates as follows. At each service completion in the i th queue at which the queue is not empty, the server makes a random decision: with probability p i (0 ⩽ pp i ⩽ 1), it serves the next customer; with probability 1 − p i , it switches to the next queue. We establish a pseudo-conservation law for this discipline with the help of a work decomposition result obtained by Boxma and Groenendijk. This result states that the amount of work in a general polling system can be decomposed into two components, only one of which depends on the service discipline. In the course of obtaining this component for the Bernoulli schedule, we establish other exact results which are of independent interest. We obtain an exact formula for the Laplace-Stieltjes transform (LST) of the steady-state waiting time associated with a particular queue in terms of the probability generating function (PGF) of the steady-state number of customers at the beginning of a service period in that queue. We also derive an expression for the joint LST-PGF of the length of a service period in a particular queue and the number of customers served during that period in terms of the PGF of the number of customers at the beginning of the service period. Making use of the pseudo-conservation law, we obtain an exact mean waiting time formula for the case where all the queues are identical.

For the M/G/ c queue we present an approximate analysis of the waiting time distribution. The result is given in the form of the defective renewal equation. This integral equation can be numerically solved by a simple recursion procedure. Also, asymptotic results for the waiting times are presented. Numerical results indicate that the approximations are sufficiently accurate for practical purposes.