Abstract
There are many queuing systems that accept single arrivals, accumulate them and service only as a group. Examples of such systems exist in various areas of human life, from traffic of transport to processing requests on a computer network. Therefore, our study is actual. In this paper some class of finite Markovian queueing models with single arrivals and group services are studied. We considered the forward Kolmogorov system for corresponding class of Markov chains. The method of obtaining bounds of convergence on the rate via the notion of the logarithmic norm of a linear operator function is not applicable here. This approach gives sharp bounds for the situation of essentially non-negative matrix of the corresponding system, but in our case it does not hold. Here we use the method of differential inequalities to obtaining bounds on the rate of convergence to the limiting characteristics for the class of finite Markovian queueing models. We obtain bounds on the rate of convergence and compute the limiting characteristics for a specific non-stationary model too. Note the results can be successfully applied for modeling complex biological systems with possible single births and deaths of a group of particles.
Highlights
Consider a Markovian queueing model on the finite state space {0, 1, ... , N } with single arrivals and group services, see the first motivation in [1] and more recent studies in [2], [3].Let X(t) be the corresponding queue-length process for any t ⩾ 0
We suppose that aij(t) = 0 for i > j − 1, all rates service do not depend on the size of a queue, i.e. ai,i+k(t) = bk(t) for k ⩾ 1, arrival rates ai,i−1(t) = λi(t)
Let us remark that the matrix B∗(t) is not essentially non-negative
Summary
Consider a Markovian queueing model on the finite state space {0, 1, ... , N } with single arrivals and group services, see the first motivation in [1] and more recent studies in [2], [3]. Consider a Markovian queueing model on the finite state space {0, 1, ... N } with single arrivals and group services, see the first motivation in [1] and more recent studies in [2], [3]. Let X(t) be the corresponding queue-length process for any t ⩾ 0. Dt where A(t) = QT(t) is the transposed intensity matrix. All column sums of this matrix are zeros for any t ⩾ 0, and A(t) is essentially nonnegative (i.e. all its off-diagonal elements are nonnegative for any t ⩾ 0), and all ‘intensity functions’ aij(t) are analytical in t. We suppose that aij(t) = 0 for i > j − 1, all rates service do not depend on the size of a queue, i.e. ai,i+k(t) = bk(t) for k ⩾ 1, arrival rates ai,i−1(t) = λi(t). BN−1(t) bN−2(t) bN−3(t) bN−4(t) aN −1N −1 (t) bbbNNNbbN1−−−(⋮(123tt((())ttt)))⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
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