Heavy traffic analysis of a queue with batch MMAP.pdf In this paper we consider a single-server work conserving queue with FIFO discipline and the following characteristics. The arrival process is a multiple batch arrival flows of customers governed by a continuous time finite state Markov chain. The service time distributions of customers may be different for arrival flows. This model was considered for the first time in [1]. The most popular Markovian arrival flows of customers is introduced in [2]. There are two extensions of arrival process. One is batch introduced in [3] that allows batch arrivals and the other is marked introduced in [4] that explicitly represents possibly correlated process considered in this paper and in paper [1] is a further extension of MAP introduced in [2 -4]. We assume that the service time distributions of customers from respective arrival streams are different from one another. In this paper we consider asymptotic behavior of a stationary distribution of the virtual waiting time and the state of a random environment under a heavy traffic. The asymptotic distribution of a stationary distribution of a virtual waiting time is an exponential with the mean depending on the parameters of random environment. In the present study we prove a heavy traffic limit theorem for steady state virtual waiting time for a model considered in [1]. We used analytical approach introduced in [6] which allows for a rigorous mathematical analysis of the stationary characteristics under a heavy traffic. The remainder of the paper is organized as follows. In section [2] we describe the model introduced in [1]. In section 3 we consider some preliminary results. Section 4 is devoted to main results. In section 4 we find the mean value of the virtual waiting time in a steady state, and there we show that the distribution of the environment becomes independent of the virtual waiting time in a steady state under heavy traffic conditions. The limit distribution of the virtual waiting time under heavy traffic conditions is an exponential distribution. 1. Model description In this paper we consider a single sever queue with the following characteristics. Arrivals to the system are from К arrival streams. We call customers from the k-th (/с = 1.2.•■ ■. AT) arrival stream class к customers. Customer arrivals are governed by a continuous-time Markov chain Z(t). We assume that Markov chain Z(t) has finite state space S = {0,1,2, • ■ -,M and is irreducible. The underlying Markov chain stais in state i e S for an exponential interval of time with a mean value ц, 1. When the sojourn time in state i has elapsed with probability g, j (0) the Markov chain Z(t) changes its state i to state j without arrivals. The chain Z(t) changes its state i to state j and n customers of class k arrive simultaneously with probability n = 1,2,---) for convenience, let cji;i=0 for all zeS. Then for all izS j=M k=K x X (aij(0) +XXCT*,

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