The queue length probability generating function for a preemptive resume priority queue characterized by Poisson arrivals and general service time distributions has been obtained by using the “supplemetary vanable method”. The preempted item follows the “resume” rule so that upon re-entry the service on the nonpriority unit is started at the point where it was interrupted when preemption occurred. Apart from the steady-state solution, the Laplace transform of the time dependent probability generating function and the length of busy periods has also been obtained.
A slight modification of the existing technique used by Gaver and by Luchak to solve queuing problems with Poisson input and a wide class of service time distributions has been made to obtain time-dependent solution of the bulk-service queuing problem. Some known results have been deduced as corollaries and the advantage of introducing this modification has been pointed out.
Summary
The bulk‐service queueing problem solved by Bailey (1954) has been extended to the case in which the maximum number of units to be taken for service is not constant, but depends upon the number of units already present with the server as well as upon the capacity. This extension has been made to study queues at lifts, bus stops, etc. The problem is first solved by applying the phase method used by Jaiswal (1960a) to solve Bailey's problem, and then by the imbedded Markov chain technique. A comparison of the two methods has been made at the end.
Summary
The method of supplementary variables has been used to obtain the Laplace transform of the time‐dependent probabilities in a head‐of‐the‐line priority queue characterized by Poisson arrivals and general service‐time distributions. An explicit solution has, however, been obtained under exponential service‐time distributions with equal mean rates. The queue length probabilities, under steady‐state conditions, have been evaluated and are found to be different from those obtained by Miller, thus showing that for complex queues, the asymptotic probability distribution obtained by considering the evolution of the queueing process in continuous time need not be the same as the one obtained by the imbedded Markov chain technique.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.