Bruneel, Herwig

Claeys, Dieter

Mélange, Willem

Walraevens, Joris

English

Ghent University Academic Bibliography

A discrete-time queueing model with constantservice times and blockingH.Bruneel, W. Mélange, D. Claeys, J. WalraevensGhent University, Departement of Telecommunications and Information ProcessingE-mail : hb,wmelange,dclaeys,jw@telin.Ugent.beWe analyze a discrete-time queueing model where two types of customers, eachhaving their own dedicated server, are accommodated in one single FCFS queue.Service times are deterministically equal to s > 1 time slots each. Customers areserved in their order of arrival, regardless of the class they belong to. We call thisservice discipline “global FCFS”. New customers enter the system according to ageneral independent arrival process, but types of consecutive customers may benon-independent in a first-order Markovian way. Specifically, we denote by α theprobability that the next customer has the same type as the previous one, andby 1 − α the probability that the next customer belongs to the opposite type asthe previous one. The parameter α can then be considered as a measure of thedegree of class clustering in the arrival process, and is referred to as the “clusterparameter”.The motivation for this work are some every day situations where this kind ofblocking occurs. A first instance are traffic jams. One reason for traffic jams aretraffic junctions. Consider for instance the following situation : vehicles approacha junction with two possible destinations (1 and 2). In the case where there is onlyone lane on the main road, it is possible that vehicles on that road heading fordestination 1 may be hindered or even blocked by vehicles heading for destination2, even when the subroad leading to destination 1 is free, simply because carsthat go to 2 are in front of them. In other words, there is a first-come-first-serve (FCFS) order on the main road regardless of the destination. Similarly, inswitching nodes of telecommunication networks, information packets with a givendestination of node 1 may have to wait for the transmission of packets destinedto node 2 that arrived earlier, even when the link to node 1 is free, if the arrivingpackets are accommodated in so-called input queues according to the source fromwhich they originate (the well-known HOL-blocking effect).We use the generating function method to calculate the main performancemeasures of the system, which are the mean system content E[u], the meansystem delay E[d] and the mean unfinished work E[w]. We derive the probabilitygenerating functions of the system content and unfinished work. By applying (thediscrete-time version of) Little’s law, the mean system delay of a customer canbe obtained.The results confirm the strong impact of “class clustering” in the arrival streamon the stability and the main performance measures of the system. For example,the form of the stability condition is given byλs < 2− α1where λ is the mean (per slot) arrival rate. This form reveals that the maximumachievable throughput of the system, expressed in work units per slot, is verydirectly determined by the degree of class clustering in the arrival process asdescribed by the cluster parameter α. Specifically, when α varies from 0 to 1, themaximum throughput is divided by a factor 2.The results also illustrate the effects of the lengths of the service times andthe burstiness in the customer arrival process. This model has some interestingscaling properties.2

A discrete-time queueing model with constant service times and blocking

https://biblio.ugent.be/publication/4260482/file/4260511

A discrete-time queueing model with constant service times and blocking

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography