A Markov modulated multi-server queue with negative customers – The MM CPP/GE/c/L G-queue

Abstract

We obtain the queue length probability distribution at equilibrium for a multi-server queue with generalised exponential service time distribution and either finite or infinite waiting room. This system is modulated by a continuous time Markov phase process. In each phase, the arrivals are a superposition of a positive and a negative arrival stream, each of which is a compound Poisson process with phase dependent parameters, i.e. a Poisson point process with bulk arrivals having geometrically distributed batch size. Such a queueing system is well suited to B-ISDN/ATM networks since it can account for both burstiness and correlation in traffic. The result is exact and is derived using the method of spectral expansion applied to the two dimensional (queue length by phase) Markov process that describes the dynamics of the system. Several variants of the system are considered, applicable to different modelling situations, such as server breakdowns, cell losses and load balancing.We also consider the departure process and derive its batch size distribution and the Laplace transform of the interdeparture time probability density function. From this, a recurrence formula is obtained for its moments. The analysis therefore provides the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes.

Generalized QBD Processes, Spectral Expansion and Performance Modeling Applications

“…where the coefficient matrices Q k can be obtained from system parameters, following the methodology in [6,8,14]. Therefore, when L is finite, the probability invariant vector v j is given by [5,6,39] …”

“…QBD-U processes are very useful in performance modeling of NGN as we shall see in the rest of this paper. Only in certain special cases of the QBD-U processes, there have been efficient, exact steady state solution methods [14,15].…”

“…This is appropriate for modeling server breakdowns, where a customer in service will be lost for sure and may be also a portion of queue of waiting customers. The RCH killing discipline is already applied to the case of the MM CPP/GE/c/1 G-queue in [14].…”

“…The CPP 3 , defined in [19] and employed in [14,29] and the K k=1 CP P k (superposition of K independent CPPs) traffic models often give a good representation of the burstiness (batch size distribution) of the traffic from one or more sources (along with mathematical tractability), but not the auto-correlations of the inter-arrival times (of batches) observed in real traffic. The usefulness and applicability of these models has been validated by measurements for example in [8,23].…”

M/M/1 retrial queue with working vacations and negative customer arrivals

“…Discrete-time queues were analyzed in [27,28,29]. Queues with negative customers have used extensively to model breakdowns, packet losses, task terminations in speculative parallelism, faulty components in manufacturing systems, server breakdowns and a reaction network of interacting molecules [30,31,32,33,34,35], Optical Burst/Packet (OBS) Switching networks [36], wireless networks [33,37,38], failures in manufacturing cells [39]. The bibliography on G-networks and negative customers can be found in [40].…”