Busy Period Processes Research Articles

New fluctuation analysis of D-policy bulk queues with multiple vacations

This paper offers new studies of the queueing process in D-policy models with Poisson bulk input, general service time, and multiple vacations. The D-policy applies when the system, after being exhausted, suspends service until the total workload crosses D (≥ 0), thereby accumulating some number of customers. The key investigation is that of the queueing process during the “first service cycle”, which includes the first vacation period when all customers line up waiting for the server to return and a period when all those customers are processed. For this model, neither the servicing process is renewal, nor the queueing process is semiregenerative relative to the departure epochs. To overcome the problem, the paper explores a new fluctuation technique of multivariate marked counting processes. It includes the time dependent analysis of queueing and busy period processes, specially developed for this process, and it yields their stationary distributions in closed analytic forms.

A continuous approximation for batch arrival queues with threshold

In this paper, the system size of batch arrival queues with threshold is approximated by a diffusion process and the accuracy is evaluated. Firstly, the busy period process is approximated by a diffusion process with an instantaneous return boundary. Secondly, the idle period process is approximated by applying the renewal theory. Then we combine the two approximations to derive the system size distribution at an arbitrary time point. The accuracy of the approximation is evaluated for a number of interarrival and service time distributions.

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated EK/G/1 queueing system as a special case.

Queues with delayed feedback

Queues with delayed feedback have been little studied in queueing theory. Presented here is a rather complete discussion of such problems including queue length processes, busy period processes and several customer flow processes (e.g., departure processes). The case in which the delay mechanism is an M-server queue is studied in detail but it is shown later that many of the results carry over to a more general delay mechanism.

Queues with delayed feedback

Queues with delayed feedback have been little studied in queueing theory. Presented here is a rather complete discussion of such problems including queue length processes, busy period processes and several customer flow processes (e.g., departure processes). The case in which the delay mechanism is an M-server queue is studied in detail but it is shown later that many of the results carry over to a more general delay mechanism.

Busy period of queues with state dependent arrival and service rates

A technique is derived providing for the analysis of busy period processes in the queueing models M/Mn /1 and Mn /M/1; queueing models with state dependent service and arrival rates respectively. A single treatment covers both models and no separate analysis is required.

Busy period of queues with state dependent arrival and service rates

A technique is derived providing for the analysis of busy period processes in the queueing models M/Mn/1 and Mn/M/1; queueing models with state dependent service and arrival rates respectively. A single treatment covers both models and no separate analysis is required.

Balking and reneging in M/G/1 systems with post-ponable interruptions

This paper considers the effect of postponable interruptions on aM/G/1 queueing process where customers balk with probabilityn/N (n=0, 1, 2, …,N) and renege after having waited for a random length of time. The busy period process in which transitions from any state (m, n) to any other state are permitted without an intermediate passage to the empty state, is investigated employing the supplementary variable method and discrete transforms. Later, the general process in which busy periods alternate with idle periods has been discussed in terms of the busy period process and renewal distributions and finally the ergodic properties of the general process studied by appealing to some results of renewal theory.

Long-run availability of paralleled systems

This paper gives an analysis of a system withN 1 components of one type andN 2 components of another type in parallel with uninterrupted running times of each component following exponential distribution and the repair time following arbitrary distribution. The head of the line and preemptive resume priority disciplines have been imposed on the repair of the components. The busy period process is investigated first using the supplementary variable method. Then the general time dependent process in which busy periods alternate with idle periods has been studied in terms of the busy period probabilities and probability of finding the repair facility idle at timet. Finally, the long-run availability of the system in terms of the steady state probabilities for the two cases of priority disciplines has been obtained.

Queuing with balking and reneging in M|G|1 systems

AM|G|1 queuing process in which units balk with a constant probability (1−β) and renege according to a negative exponential distribution has been considered. The busy period process is first investigated making use of the supplementary variable technique and discrete transforms. The expression for the joint distribution of the number of customers serviced during a busy period and the length of the busy period has been derived. FollowingGaver (1959) the general process is investigated and making use of renewal theory the ergodic properties of the general process have been studied. It has been shown that as long as reneging is permitted (α>0), the steady states always exist, but when no reneging is permitted (α=0), the steady states exist only whenλ β η<1.

Queues with semi-Markovian arrivals

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

Queues with semi-Markovian arrivals

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

THE EFFECT OF POSTPONABLE INTERRUPTIONS ON A M/G/1 SYSTEM WITH IMPATIENT CUSTOMERS

AbstractThis paper considers a waiting line system where units become impatient after having waited for certain time and leave the queue (renege) without being serviced. The servicing of the units is subject to interruption by the arrival of an “interruption ” possessing a priority for service over the ordinary units, head‐of‐the‐line priority discipline being prevalent. The busy period process is investigated first, making use of the supplementary variable method. Later, the general process is studied in terms of the busy period process and renewal distributions. Lastly, the ergodic properties of the general process are examined by appealing to some results of renewal theory.