Thoddi C. T. Kotiah

Approximate Transient Analysis of Some Queuing Systems

We propose approximations for the time-dependent queue size distribution Pnt and the mean queue size E[Xt] of an M/M/1 queue. With a Laplace transform-generating function technique, the exact analysis involves the complex root u* inside the unit circle of a certain quadratic. The exact value of u* leads to transforms that, when inverted, yield rather complicated expressions for Pnt and E[Xt]. We discuss three iterative procedures that give successive rational approximations for u*. These approximations yield, for the queue size distribution and the mean queue size, approximate transforms that are rational and easier to invert. We obtain sharp bounds for Pnt and E[Xt] for a wide range of parameter values by performing the second iteration and combining the approximate results given by the iterative procedures. The methods can be applied to approximate the behavior of more complicated queues. Applications to the M/M/2 queue and the M/Ep/1 queue are discussed briefly.

Occurrences of cycling and other phenomena arising in a class of linear programming models

An investigation into the average queue size for a certain class of queues has resulted in the formulation of linear programming problems which are ill-conditioned in some cases. In attempting to solve these linear programming models, using IBMs MPS package, instances of cycling were encountered. Small perturbations in the input data resulted in problems which did not cycle. This fact, plus several other observed phenomena suggest that the primary reason that cycling is not known to occur more frequently is that round-off errors in the computations perturb the problem sufficiently to prevent cycling (or at least to prevent indefinite cycling). In one case maximizing and minimizing an objective function subject to the same constraint set was attempted, but MPS solved only one of these while giving an indication of infeasibility for the other.

On Two-Server Poisson Queues with Two Types of Customers

This paper considers two schemes for a steady-state queue with two types of customers served by two desks, the customers arriving mixed in a Poisson stream and having negative-exponential service-time distributions, with different means characterizing the two types of customers. In Scheme I each type has a particular desk, for which he queues on arrival. In Scheme II all customers keep in a single queue and proceed indifferently to either desk. Numerical remits indicate that Scheme II reduces mean queue size and queuing times, generally with advantage to both types of customers.

Sums of powers of integers—A review

The problem of finding formulae for the sums of powers of integers has been tackled by various methods throughout the literature. A review of the main techniques is given with emphasis on both the expanded polynomial forms and the more familiar compact factored forms of the formulae. For large powers, the factored forms can be obtained more readily by first deriving a special well‐defined polynomial form called the Faulhaber form. Most of the material on algorithms for generating the coefficients of the Falhauber polynomials is new. The expanded forms of the formulae involve Bernoulli numbers which are important in other applications of interest to mathematics and statistics majors.

On a Linear Programming Technique for the Steady-State Behavior of Some Queuing Systems

We consider the class of stationary queuing processes whose behavior can be described by a system of difference equations that are linear in the probabilities of the states. We describe a linear programming technique that will provide bounds of any prescribed degree of accuracy for any steady-state quantity expressible linearly in terms of the probabilities. For some queues a standard technique such as the use of generating functions may require a lengthy analysis and may not provide a check on the results. In contrast, the proposed technique is always easy to implement on a computer and, in a certain sense, is self-checking. Illustrative examples are given. The technique is implemented through an algorithm that, as shown by the examples, may give useful results after only a few iterations. We prove that the probability of zero delay for the M/D/2 queue is larger than that for an M/M/2 queue with the same traffic intensity. For a large class of parameter values of the multi-server mixed queue we obtain, with very little computing time, sharp bounds for the average queue size.

The use of linear programming in some common applications of Bayes’ rule

Bayes’ theorem is often applied in problems that essentially involve finding the accuracy of atesting device, given the sensitivity of the device. It is shown how linear programming can be used to determine (in a certain sense) the minimum sensitivity needed for a given prescribed minimum accuracy.