Stationary distribution invarianceof an open queueing network with temporarily non-active customers

This paper considers stationary functioning of an open queueing network with temporarilynon-active customers. Systems set is J = {1,…,N}. Non-active customers are located in a systemqueue and do not get service. Customers may turn from non-active state into state, when they mayget their service and vice versa; ni(t), ni(t) denote the numbers of ordinary and non-active customersat system iJ at time t accordingly. Service times are assumed to be independent randomdistributed values with distribution functions ( ' , , ') Bi ni +ni xini+ni and mean values 1/ ƒ i , iJ.Under conditions of ergodicity: ƒi<ƒi, ƒiƒi<ƒiƒi, iJ, process z(t) = ((n1(t),n1(t)),…, (nN(t),nN(t)))has stationary distribution {p(z), zZ}, which may be founded by formulas:p((n1,n'1),...,(nN,n'N)) = p1(n1,n'1)...pN(nN,n'N) ,where'( , ' )ni nii iii i ii iip n n⎛ƒ ⎞ ⎛ƒƒ ⎞= ⎜⎝ƒ ⎟⎠ ⎜⎝ƒϕ ⎟⎠; here ƒi is an unique positive solution to the system of trafficequations, iJ. Stationary distribution which is invariant to service time distribution is obtained.

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