Computation of the moments of queue length in the queue
Introduction
The queue as a single server system with infinite buffer, Batch Markovian Arrival Process () and independent identically arbitrarily distributed service time is very important for applications queueing model because the (see [2], [11]) is an adequate mathematical model of bursty, correlated flows of information in modern telecommunication networks. Therefore its analysis was very important theoretical task. Initially, this task was solved long time ago by V. Ramaswami in [14] where essentially the same arrival process as the was called as process. The queue was then analysed in [11]. However, until now analysis of such a system attracts attention of researchers, see, e.g., very recent paper [17].
As it was mentioned above, the type model assumes independence and identical distribution of service times of successive customers. In many real-world systems, service times of successive customers may be dependent and have different distributions. To take into account such a dependence, so called Semi-Markovian () service process was considered, see, e.g., [3], [18]. Importance of consideration of queues with service process for practical needs is stressed, e.g., in recent work [1] and references therein. In the paper [16], a very general model of the type with possible dependence of inter-arrival and service times was considered under assumption that the marginal distribution of service times is of phase type. The type queue without such an assumption and with batch arrivals was analysed in [7], [12]. The problem of computation of steady state distribution of queue length and waiting time distribution in the system was successfully solved. In this communication, we supplement results of [12] by the effective recursive procedures for computation of the moments of the queue length distributions. Moments have an important role for performance evaluation of various queueing systems. Sometimes, information about the mean value and variance of queue length is enough for managerial decisions. If this information is insufficient and the shape of queue length distribution is of a primary interest (e.g., to evaluate the probability that the queue length will exceed a certain important level) while this shape hardly can be found exactly, the shape can be estimated numerically based on the knowledge of the value of several moments of the distribution, see, e.g. [19]. Effective recursive procedures for computing the moments of the queue length at service completion epochs and arbitrary time for queue were given in [6]. Direct extension of results from [6] appears not possible for the type queue because that results essentially exploit irreducibility of some matrix generating functions at the point while they are reducible when the service is of type. In this communication we elaborate the recursive procedures for computation of the moments of the queue length in this type system.
Section snippets
Preliminary results
We consider a single server system with an infinite buffer. The arrival process is the It is defined by the underlying process which is an irreducible continuous time Markov chain with a finite state space and with the matrix generating function of square matrices of size consisting of the intensities of transitions of the Markov chain accompanied by the generation of -size batch of customers, The matrix is an infinitesimal
The recursive procedure for computing the moments of distribution of the embedded Markov chain
For brevity of presentation, let us denote Then, Eq. (2) can be rewritten in the form
Because the matrix is assumed to be irreducible, it is easy to verify that the matrix is irreducible as well. This allows us to calculate the vector factorial moments using the slight modification of the recursive procedure presented in [6].
Theorem 1 Let
Then the vector factorial moments
The recursive procedure for computing the moments of the queue length distribution at an arbitrary time
Denoting Eq. (3) is easily rewritten in the form
Theorem 2 Let
Then the vector factorial moments of the system states distribution at an arbitrary time
are computed recursively by
where
Conclusion
The problem of computing the factorial moments of queue length distribution in the quite general queueing system of type is considered. This queueing model allows to effectively take into account possible correlation and shape of distributions of inter-arrival and service times what is very important in some real world systems. Algorithms for computation of the moments are based on the use of the known vector functional equation for the vector generating function of queue length
Acknowledgement
The publication was financially supported by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.a03.21.0008).
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