On the rate of convergence for a class of Markovian queues with group services

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.


Introduction
Consider a Markovian queueing model on the finite state space {0, 1, … , } with single arrivals and group services, see the first motivation in [1] and more recent studies in [2], [3].
Let ( ) be the corresponding queue-length process for any ⩾ 0. The probabilistic dynamics of the process ( ) is described by the forward Kolmogorov system where ( ) = ( ) is the transposed intensity matrix. All column sums of this matrix are zeros for any ⩾ 0, and ( ) is essentially nonnegative (i.e. all its off-diagonal elements are nonnegative for any ⩾ 0), and all 'intensity functions' ( ) are analytical in . We suppose that ( ) = 0 for > − 1, all rates service do not depend on the size of a queue, i.e. , + ( ) = ( ) for ⩾ 1, arrival rates , −1 ( ) = ( ). The process ( ) belongs to class (III), see [3]. The matrix ( ) for ( ) has the following structure:

Stationary Markovian queueing model
In this paper we consider a subclass of the class (III) satisfying additional suppositions ( ) = 0, 1 ⩽ ⩽ − 1, ( ) = ( ) and ( ) = ( ) for any , ⩾ 0. The difficulty of studying this model is due to the fact that it is not possible to apply the most convenient method of the logarithmic norm for it, see [3]. Now we get the following expression for the transposed intensity matrix: We perform the following system transformations. Since 0 ( ) = 1 − ∑ ⩾1 ( ), one can rewrite the system (1) as All bounds on the rate of convergence to the limiting regime for ( ) correspond to the same bounds of the solutions of system Denote by upper triangular matrix Let u( ) = y( ), then Let us remark that the matrix * ( ) is not essentially non-negative. This means that the method of the logarithmic norm is inconvenient to apply (it gives poor results). That's why we use the method of 'differential inequalities', which was described in [4]- [6].
Let D = diag ( 1 , 2 , … ) be a diagonal matrix and , = 1, … , be nonzero numbers. By w( ) denote a product Du( ), then one can rewrite (8) as following system 208 DCM&ACS. 2020, 28 (3) 205-215 By u( ) denote an arbitrary solution of system (8), then we can consider an interval ( 1 , 2 ) with fixed signs of the coordinates ( ) and choose the elements of the diagonal matrix such that signs of the entries are equal with signs of corresponding coordinates ( ) of the solution of system (8).
Since any ( ) ⩾ 0 on the corresponding time interval, the sum If we compared all the norms, then one can obtain the final bound where is a corresponding constant. In our case (in general, all intensities depend on the time ) and we have = ⋅ (1 − ℎ). 2) Let all 1 , …, be positive, and all +1 , …, negative. Similarly be positive, and all +1 , …, negative and all +1 , …, be positive too. As before | | > | +1 |.
The right-hand side of estimate (16) decreases rather slowly with increasing . However, this does not mean that the estimate is inaccurate because the real rate of convergence is rather slow. For example, let = 50 and ℎ = 1 2 . In Figures 1-3 we can see the state probability of an empty queue 0 ( ) as low convergence rate function of time .

Conclusions
Some new class of finite Markovian queueing models with single arrivals and group services was considered. Bounds on the rate of convergence for these models and computations of the limiting characteristics for a specific non-stationary model were obtained.
The obtained results belong to the theory of queueing systems and can be applied, for example, in medical and biological stochastic systems, which satisfy the adopted assumptions.
For describing possibility of applications of Markovian queues we can refer to [7]- [17], which contains a broad overview and a classification of timedependent queueing systems.