The MM ∑ K k = 1 CPP k / GE / c / L G-Queue as a Computable Node Model : Steady State Solution , Applications and Extensions

This is a homogeneous multi-server FCFS queue with finite or infinite queueing capacity. The service time distribution at a server is generalised exponential. The arrival process is the superposition of K independent compound Poisson processes ( CPPk ; k = 1, 2, . . . ,K) of (positive) customers and a CPP of negative customers. All the arrival and the service processes are jointly modulatedby anN -state Markov phase process. Exact solutions to the steady state joint probabilities of queue length by phase are derived. The use of negative customers (with the RCE-immune servicingkilling discipline or another) is to provide a platform for many possible studies including load-balancing and the effect of server-breakdowns. This complex queue, capable of representing correlations and burstiness of incoming traffic, capable of handling large batch sizes of geometric or non-geometric distribution in both arrivals and services, with its viable and efficient steady-state solution, can provide a basis for finding an appropriate as well as computable model for nodes in optical networks. I. I NTRODUCTION A number of models have been proposed for describing the traffic that arises in today’s telecommunications systems such as UMTS and IP networks, which often exhibits burstiness – i.e. batches of transmission units (e.g. packets) arrive together – and correlation among interarrival times. These models include the compound Poisson process (CPP) in which the interarrival times are assumed to have generalised exponential (GE) probability distribution [1], the Markov modulated Poisson process (MMPP) and self-similar traffic models such as Fractional Brownian Motion (FBM) [2], [3]. A CPP traffic model often gives a good representation of burstiness of the traffic from one or more sources, e.g. [4], [5], but not of the auto-correlations observed in much real traffic. Conversely, the MMPP models can capture auto-correlation but not burstiness, e.g. [6]–[8]. The self-similar models such as FBM can account for both auto-correlation and burstiness, but they are analytically intractable in a queueing context. Often, the traffic to a node is the superposition of traffic from a number of sources complicating the system analysis further. The earlier models, viz. the MMCPP/GE/c/L queue, the MM CPP/GE/c/L GQueue introduced by us, have been capable of accommodating large batch sizes of arrivals and services with an efficient solution in the Markovian framework. However, they suffer from the restriction on the batch size distribution. In order to relax that restriction, to make the model vastly flexible and also to accommodate superposition of many arrival streams, we introduce a new queueing/traffic system, the Markov Modulated ∑K k=1CPPk/GE/c/L G-queue. This is a homogeneous multiserver queue withc servers, GE service times and with K independent positive and an independent negative customer arrival streams, each of which is a CPP, i.e. a Poisson point process with batch arrivals of geometrically distributed batch size. In other words, interarrival times of each of the K + 1 arrival streams individually are also independent GE random variables. All the K + 2 GE distributions (for positive and negative customer interarrival times and for service time) are jointly modulated by a continuous time Markov phase process (CTMP), also termed Ram Chakka is with Department of Computer Science, Norfolk State University, 700 Park Avenue, Norfolk, VA 23504, USA. Tel: +1 757 823 2163, Email: ramchakka@yahoo.com Tien Van Do is with Department of Telecommunications, Budapest University of Technology and Economics H-1111, Magyar tud ósok körútja 2. , Budapest, Hungary Tel: +36 1 4632070, Email: do@hit.bme. hu (Correspondence author ) Zsolt Ṕandi is with Department of Telecommunications, Budapest University of Technology and Economics H-1111, Magyar tud ósok körútja 2. , Budapest, Hungary Tel: +36 1 4632070, Email: pandi@hit.bme. hu