This paper considers an M/G/1 queue where the service time for each customer is a discrete random variable taking one of N values. We call this an M/DN/1 queue. There are potential numerical problems inverting Laplace transforms associated with this queue because the service distribution is discontinuous. The purpose of this paper is to investigate the performance of numerical transform-inversion methods in analyzing this queue. We first derive continuity properties of the steady-state distribution of wait in the M/DN/1 queue. Then, we show analytically how continuity properties affect the performance of the Fourier method for inverting transforms. In particular, continuity is not required in all derivatives for best performance of the method. We also present a new inversion method specifically for the M/DN/1 queue. Finally, we give numerical experiments comparing these and four other inversion methods. Although no method clearly dominates, the recursion method performs well in most examples.

In many modern applications of queueing theory, the classical assumption of exponentially decaying service distributions does not apply. In particular, Internet and insurance risk problems may involve heavy-tailed distributions. A difficulty with heavy-tailed distributions is that they may not have closed-form, analytic Laplace transforms. This makes numerical methods, which use the Laplace transform, challenging. In this paper, we develop a method for approximating Laplace transforms. Using the approximation, we give algorithms to compute the steady state probability distribution of the waiting time of an M/G/1 queue to a desired accuracy. We give several numerical examples, and we validate the approximation with known results where possible or with simulations otherwise. We also give convergence proofs for the methods.

Internet traffic flows have often been characterized as having power-tailed (long-tailed, fat-tailed, heavy-tailed) packet interarrival times or service requirements. In this work, we focus on the development of complete and computationally efficient steady-state solutions of queues with power-tailed interarrival times when the service times are assumed exponential. The classical method for obtaining the steady-state probabilities and delay-time distributions for the G/M/1 (G/M/c) queue requires solving a root-finding problem involving the Laplace-Stieltjes transform of the interarrival-time distribution function. Then the exponential service assumption is combined with the derived geometric arrival-point probabilities to get both the limiting general-time state and delay distributions. However, in situations where there is a power tail, the interarrival transform is typically quite complicated and never analytically tractable. In addition, it is possible that there is only a degenerate steady-state system-size probability distribution. Thus, an alternative approach to obtaining a steady-state solution is typically needed when power-tailed interarrivals are present. We exploit the exponentiality of the steady-state delay distributions for the G/M/1 and G/M/c queues, using level-crossings and a transform approximation method, to develop an alternative root-finding problem when there are power-tailed interarrival times. Extensive computational results are given.

In this paper we present an enhanced extension to Wilkinson's equivalent random technique that allows one to easily compute the different levels of blocking that various streams of traffic see using the same trunk group. This empirical extension is based on a regression analysis and is superior to a similar extension by Katz. An application of our extension to system sizing problems in AUTOVON is presented. Previous sizing methods in the AUTOVON access area did not properly account for the peakedness of overflow traffic and thereby violated desired grades of service. The methods presented here overcome this problem, and yield a lower cost solution using fewer two-way trunks.

In this paper a mathematical model is developed for the performance of crisis calls. In such a system, a finite number of crisis calls attempt to use a communication link that has been receiving ordinary calls. The performance of the link with respect to the crisis calls is analyzed and studied. A numerical analysis is conducted to determine the behavior of the crisis calls under various scenarios. Some simple upper and lower bounds on the blocking and loss probabilities for crisis calls are given.

This paper describes the results of a study to calculate the grade of service (GOS) in a telephone network carrying two classes of traffic. When blocked, the higher class calls may preempt calls of the lower class. Two preemption disciplines are considered in this study. They are the ruthless and friendly search preemption disciplines of channels on interswitch trunk groups. The results of a computer model agree very well with a proven network simulator. The CPU time to determine the network GOS was reduced from hours to minutes.

In this paper we consider a system where the talkspurts from several voice conversations are buffered and multiplexed over the same transmission channels. A simple mathematical model is given for the case where the number of voice calls may statistically come and go. This model is then used to numerically analyze the system and make comparisons with previously presented results.

We develop a mathematical analysis for the steady state performance of a system where voice calls and data packets are transmitted over the same channel. The voice calls have priority over the data packets, in that the data packets are transmitted only when there are no voice calls present in the system or the voice conversation is in a long silent period.

Recent papers have introduced a multiplexing structure for mixing voice and data traffic in an integrated telecommunications system. This structure utilizes a master frame format of a time division statistical multiplex facility. A certain portion of the frame is allocated to voice calls, and data traffic is assigned to the remaining frame capacity. To achieve a high transmission utilization, data are allowed to use any residual voice capacity momentarily available due to statistical variations in the voice traffic. The voice traffic is treated as a loss system and data packets are buffered. In this paper we derive exact analytical expressions for the key system perfomance measures, the probability of loss for voice calls, and the expected waiting time for data packets. Actually, two cases are considered, the one discussed above, called the movable boundary case, and one where the boundary is fixed; i.e., data are not allowed to utilize the residual voice capacity. The computational aspects of calculating actual numbers are discussed in some detail, and results are presented for typical cases.