M/G/1 Queuing Theory and Applications

Abstract

The M/G/1 theory is a powerful tool, generalizing the solution of Markovian queues to the case of general service time distributions. There are many applications of the M/G/1 theory in the field of telecommunications; for instance, it can be used to study the queuing of fixed-size packets to be transmitted on a given link (i.e., M/D/1 case). Moreover, this theory yields results which are compatible with the M/M/1 theory, based on birth-death Markov chains.In the M/G/1 theory, the arrival process is Poisson with mean arrival rate λ, but, in general, the service time is not exponentially distributed. Hence, the service process has a certain memory: if there is a request in service at a given instant, its residual service time has a distribution depending on the time elapsed since the beginning of its service. Let us refer to a generic instant t. The system is described by a two-dimensional state S(t), characterized as follows:• Number of requests in the system at instant t, n(t).• Elapsed time from the beginning of the service of the currently served request, τ(t). Note that in the Markovian M/M/1 case, the pdf of the residual service time does not depend on τ(t) because of the memoryless property of the exponential distribution.Hence, S(t) ¼ {n(t), τ(t)}. In order to characterize these queues, we study their behaviors at specific time instants ζ i where we obtain a mono-dimensional simplification of state S(ζ i ). The M/G/1 queue is studied at specific imbedding instants, where we obtain again a Markovian system; this is a so-called imbedded Markov chain [1,2]. Different alternatives are available to select instants ζ i . It is not requested that instants ζ i be equally spaced in time. Typical choices for ζ i instants are:The online version of this chapter (

IoT Based Real Time Traffic Monitoring System Using M/G/1 Queuing Research

“…During traffic congestion the arrival rate is the exponential but the departure or service rate is not exponential but follows general distribution which results in queue formation. M/G/1 queueing model can be used in analysing the delay in data packets in telecommunication channel [5]. In this paper we consider M/G/1 model to analyse the traffic data as it suits for traffic congestion scenario where vehicles enter exponentially but departure is not exponential and can follow general distribution if remedy is provided for congestion control.…”

Radio Resource Sharing Framework for Cooperative Multioperator Networks With Dynamic Overflow Modeling

“…We assume that the network operators consider a loss model, where there are no waiting places in the system, and it blocks the arriving channel requests when all servers are busy [28]. Unlike the queueing type models, loss models are stable and the closed form analytical solution of blocking probability exists irrespective of traffic intensity.…”

Performance Modeling of Parallel Computers NOW and Grid

“…(Tandem network), for which we can get following final solution, that the probability of k 1 demands at first node and k 2 demands at second node is The final expression proves by evidence independence of both M/M/1 queuing theory systems. Generalization to the U queuing theory systems of M/M/1 or M/M/m made Jackson (Jackson theorem) [5,9]. Several different ways of identifying this sort of behavior have been proposed, but the name product form comes from Jackson`s theorem, which expresses the joint probability of the numbers of customers at each queue being a particular combination is the product of their individual probabilities of having that number.…”

“…In case of possible solving system of linear equations (SLE) to find in analytical way node's λ i and λ ij respectively input intensities, most parallel algorithms use to its solution Gauss elimination method (GEM). These GEM parallel algorithms have computation complexity given as O(n 3 ) floating point multiplications and a similar number of additions [2,9]. These values are however adequate to handle most existing communication network of based NOW module.…”