The $MAP/PH/1/N$ queue with flows of customers as a model for traffic control in telecommunication networks

Abstract

A single-server queueing model with finite buffer and flows of customers is considered. Flow means a group of customers which should be sequentially processed in the system. In contrast to the standard batch arrival when a whole group of customers arrives into the system at one epoch, we assume that the customers of an accepted flow arrive one by one in exponentially distributed times. Service time has Phase type ($PH$) distribution. Generation of flows is described by the Markov Arrival Process ($MAP$). A flow consists of a random number of customers. This number is geometrically distributed and is not known at a flow arrival epoch. The number of flows, which can be admitted into the system simultaneously, is subject to control. Accepted flow can be lost, with a given probability, in the case of any customer from this flow rejection. Analysis of the joint distribution of the number of flows and customers in the system, flow loss probability and sojourn time distribution is implemented by means of the matrix technique and method of catastrophes. The effect of control on the main performance measures of the system is demonstrated numerically. The influence of correlation in the arrival process of flows, variation of service time and probability of a flow loss in case of any customer from this flow rejection is illustrated.

Introduction

Many problems in routing, flow control, bandwidth allocation and memory management in telecommunication networks can be solved with help of queueing theory. Typically, a user of a telecommunication network generates not a single packet but a whole bunch of packets, and service of this user assumes transmission of all of these packets. This is why the batch arrivals are often assumed in the analysis of queueing systems. It is usually assumed that, at a batch arrival epoch, all customers of this batch arrive into the system simultaneously and the decision as to whether or not the batch should be admitted into the system is based on comparison of the batch size and the available capacity of the system, see, e.g., [1], [2], [3]. A very general model of the $BMAP/SM/1/N$ type with discipline of partial admission was investigated in [4], and the $BMAP/G/1/N$ type with disciplines of complete rejection and complete admission were investigated in [5]. Numerically stable algorithms accounting for the special structure of the transition probability matrix, which are suitable even for the buffer capacity $N$ equal to several thousands, are presented there.

However, the typical feature of many current communication networks is that customers arrive in batches and arrival of customers, which belong to a given batch, is not instantaneous but is distributed in time. The first customer of a batch arrives at the batch arrival epoch while the rest of customers arrive one by one at random intervals. To distinguish such mechanism of customer arrivals from the standard batch arrivals, further we refer to such batches of customers as flows. The flow size is random and it may be not known a priori at the flow arrival epoch. Such a situation is typical, e.g., in modeling transmission of video and multimedia information. This situation is also typical in $IP$ networks. It is discussed, e.g., in [6] with respect to the modeling Scheme of Alternative Packet Overflow Routing ($SAPOR$) in $IP$ networks.

In this scheme, the flow represents a set of packets that should be sequentially routed in the same channel. When a packet arrives, it is determined (e.g. by means of $IP$ address) if the packet is part of a flow, already tracked. If the packet belongs to an existing flow, the packet is marked for transmission. If the flow is not yet tracked and the buffer and channel capacity is still available, the packet is admitted into the system and flow count is increased. Otherwise the flow is routed on the overflow link (or is dropped) and the packet is rejected in the considered channel. Tracked flows are cleared after they are finished. Clearing of inactive flow is done if no more packets belonging to this flow are received within a certain time interval. Tracking and clearing of flows is performed by a token mechanism. Physically, the token can be interpreted, e.g., as some timer which is switched-on at a flow admission epoch and is restarted at epochs of other customers from this flow arrival and is switched-off if some fixed timeout expires but a new customer from this flow does not arrive. The number of tokens (timers), which defines the maximal number of flows that can be admitted into the system simultaneously, is a very important control parameter. If this number is small, the channel may be under-utilised. If this number is too large, the channel may become congested. Many packets from admitted flows may be lost and Grade of Service becomes bad. Simultaneously, delay and jitter of flows may essentially increase. So, the problem of defining the optimal number of tokens is practically important and non-trivial. In [6], performance measures of this $SAPOR$ scheme of routing in $IP$ networks are evaluated by means of computer simulation.

Note that this scheme regulates namely flows admission into the system, not all their processing, and assumes the return of a token into the pool of available tokens at the epoch of flow arrival termination, not at the moment when all customers from this flow leave the system after the service. Alternative scheme is hardly realizable, especially when the server represents, not a concrete device (router, channel, computer, etc), but a whole virtual path in $IP$ network.

An analogous situation also naturally arises in modeling information retrieval in relational databases where, besides the CPU and disc memory, some additional “threads” or “connections” should be provided to start the user’s application processing. In this interpretation, flow means application while customers are queries to be processed within this application.

In the paper [7], the queueing model with a finite buffer, stationary Poisson arrival process of flows and exponential service time distribution, which suits performance evaluation and capacity planning of the $SAPOR$ routing scheme as well of other real life systems with time distributed arrival of customers in a flow, is considered. To the best of our knowledge, such kinds of queueing models were not considered and investigated in literature previously. In [7], the problem of the system throughput maximization, subject to restriction of loss probability for customers from accepted flows, is solved.

In the present paper, the analysis given in [7] is essentially extended in three directions. Instead of the stationary Poisson arrival process of flows, the Markov Arrival Process (MAP) is considered. It allows one to catch the effect of correlation in the arrival process of flows. The presented numerical results show that the correlation has a profound effect on the system performance measures. This is the main motivation of the research summarized in this paper. Use of the simpler model analyzed in [7] can cause huge errors in prediction of a system performance in situations when the arrival process is correlated, while correlation is the essential feature of a traffic in modern communication networks.

The second direction of extension is consideration of the Phase type (PH) service process instead of exponential service time distribution assumed in [7]. Usefulness of PH distribution in the description of service process in telecommunication networks is stated, e.g., in [8], [9]. Presented in this paper, analysis allows one to take into account variation of the service time carefully. Impact of service time variation is illustrated numerically in this paper as well.

The third direction of extension is the following one. It is assumed in [7], that the loss (due to a buffer overflow) of the customer from the accepted flow never causes loss of a whole flow itself. A more realistic assumption in some situations is that the accepted flow might be lost (terminates connection ahead of schedule), e.g., it can happen if the percentage of lost voice or video packets (and quality of speech or movie) becomes unacceptable for the user. To take such a possibility into account to some extent, in this paper we assume that the loss of a customer, with a fixed probability, leads to the loss of a flow to which this customer belongs. Influence of this probability is numerically investigated in this paper as well. Note that although this extension of a model implies only slight modification of the Markov chain under study, the problem of calculating probability of an accepted flow loss is not trivial and this problem found a nice solution in this paper.

The rest of the paper is organized as follows. In Section 2, the model is described. Steady state joint distribution of the number of flows and customers in the system is analyzed and expressions for the main performance measures of the system are given in Section 3. Section 4 is devoted to consideration of the customer and the flow sojourn time distribution. In Section 5, loss probability for arbitrary flow admitted into the system is calculated. Section 6 contains numerical illustrations and their short discussion and Section 7 concludes the paper.