STATIONARY DISTRIBUTION IN THE SYSTEM WITH THE STAYING INTENSITY OF THE INPUT FLOW

Consider a single server queuing system with an infinite number of waiting places and service intensity representing by some random process. A Markov chain with many states defines the process characterizing the intensity of the input flow. The Markov chain describes a game of player ruin and defines a piecewise constant random process with many states which is Markov process. A queuing system with so defined input flow rate will be denoted M. Consider the Markov process characterizing the random intensity of the input flow and the random number of customers in the system at a time. Then assume that the following inequalities are satisfied: Lambda(0) < mu,...,Lambda(N) < mu, Lambda(i)not equal Lambda(j), i not equal j. Our task is to calculate the stationary distribution of the process, describing a number of customers in the system M. It is obvious that the stationary distribution of the Markov chain depends on the initial state At first glance, this dependence makes it difficult to solve the problem. However, the analogy of the Markov chain process, describing the game to ruin the player, on the contrary, simplifies the computation of the stationary distribution of the process in the system M. The solution of this problem can be reducing to the solution of a discrete analogue of the Dirichlet equations. Then it is possible to use well-known formulas for stationary distribution service process in the system with a constant intensity of the input flow. Consider a Markov chain with a set of states with nonzero elements of the transition probability matrix theta(i,j+1) = p, theta(i,j-1) = q, i=1,..., N-1, theta(0,0) = theta(N,N) = 1, 0 < p < 1, q = 1 - p. (1) Everywhere further, probability P-z0(A)denotes the probability of an event A provided that the Markov chain, z(k), k = 0,1, ..., takes the initial value z(0), Denote pi(z0) (k) = Sigma P-N-1(z=1)z0 (z(k) = z), psi(z0) (k) = P-z0 (z(k) = 0) + P-z0 (z(k) = N). It is obvious that the following relation is valid pi(z0) (k) + psi(z0) (k) = 1. (2) Formulas (1), (2) are true at any z(0) = 0,...N. Denote rho(0) = Lambda(0)/mu < 1, rho(N) = Lambda(N)/mu < 1, then stationary distribution P-z0(k), k = 0,1,..., of a number of customers in the queuing system M satisfies the equality P-z0 (k) = psi(z0)(0)(1 - rho(0))rho(k)(0) + psi(z0) (N)(1 - rho(N)rho(k)(N), k = 0,1,...