Oj Onno Boxma

The Product Form for Sojourn Time Distributions in Cyclic Exponential Queues

Consider a closed cyclic queuing system consisting of M exponential queues. The Laplace-Stieltjes transform of the joint distribution of the consecutive sojourn times of a customer at the M queues is determined and shown to have a product form. The proof is based on a reversibility argument.

On response time and cycle time distributions in a two-stage cyclic queue

We consider a two-stage closed cyclic queueing model. For the case of an exponential server at each queue we derive the joint distribution of the successive response times of a custumer at both queues, using a reversibility argument. This joint distribution turns out to have a product form. The correlation coefficient is calculated and shown to be non-positive. n nFor the case of one general server and one exponential server we derive two approximations for the joint distribution of the response times. The numerical results based on these approximations are compared with simulation results. The first approximation which heavily relies on results for the M/G/1 queue with finite capacity, is somewhat complicated, but it yields exact marginal distributions and appears to be very accurate. The second one is less accurate, but very easily ???.

Joint distribution of sojourn time and queue length in the M/G/1 queue with (in) finite capacity

Abstract For the M/G/1 queue we study the joint distribution of the number of customers x present immediately before an arrival epoch and of the residual service time ζ of the customer in service at this epoch. The correlation coefficient ϱ (x, ζ) is shown to be positive (negative) when the service time distribution is DFR (IFR). The result for the joint distribution of x and ζ leads to the joint distribution of x, of the sojourn time s of the arriving customer and of the number of customers z left behind by this customer at his departure. ϱ(x, s), ϱ(z, s) and ϱ(x, z) are shown to be positive; ϱ(x, s) and ϱ(z, s) are compared in some detail. Subsequently the M/G/1 queue with finite capacity is considered; the joint distributions of x and ζ and of x and s are derived. These results may be used to study the cycle time distribution in a two-stage cyclic queue.

On the longest service time in a busy period of the M⧸G⧸1 queue

This paper considers the supremum m of the service times of the customers served in a busy period of the M[+45 degree rule]G[+45 degree rule]1 queueing system. An implicit expression for the distribution m(w) of m is derived. This expression leads to some bounds for m(w), while it can also be used to obtain numerical results. The tail behaviour of m(w) is investigated, too. The results are particularly useful in the analysis of a class of tandem queueing systems.

M/G/∞ tandem queues

We consider a series of queues with Poisson input. Each queueing system contains an infinite number of service channels. The service times in each channel have a general distribution. n nFor this M/G∞ tandem model we obtain the joint time-dependent distribution of queue length and residual service times at each queue. This leads to an expression for the joint stationary distribution of the number of customers in various queues at the arrival epochs of a tagged customer at those queues.

The longest service time in a busy period

This paper considers the supremumm of the service times of the customers served in a busy period. For theG/G/s queue the tail behaviour of the distributionm (w) ofm is compared with that of the service time distribution. For theEk/G/1 queue an expression for the joint distribution ofm and of the number of customers served in the busy period is derived.Finally some detailed results for theM/M/1 queue are mentioned.ZusammenfassungDiese Arbeit befaßt sich mit dem Supremumm der Bedienungszeiten der in einer Arbeitsperiode bedienten Kunden. Im Falle des BedienungssystemsG/G/s werden die Wahrscheinlichkeiten für große Werte vonm verglichen mit den Wahrscheinlichkeiten für große Bedienungszeiten. Im Falle vonEk/G/1 wird ein expliziter Ausdruck für die gemeinsame Verteilung vonm und der Anzahl der in der Arbeitsperiode bedienten Kunden gewonnen. Einige Resultate fürM/M/1 bilden den Abschluß.

A probabilistic analysis of multiprocessor list scheduling : the Erlang case

Let (T1,…,Tn) denote and ordered list of n tasks, which have to be served on m≥2 parallel processors. X1 denotes the service time of Ti (X1,…,Xn) is a set of independent, identically distributed st...