Investigation of queueing network with unreliable systems at transient regime

The open exponentional queuing network with unreliable systems which functioning under condition of heavy loading is investigated. The Poisson flow of rate lambda(t) enters the network. The service time of messages in each of network systems has exponential distribution with parameter mu(i) (t), i = (1, n) over bar. Service channels are exposed to random failure and serviceable work time of each channel of system Si has exponential distribution with parameter beta(i) (t), i = (1, n) over bar. After failure the service channel immediately starts to be restored and restoration time also has exponential distribution with parameter gamma(i) (t), i = (1, n) over bar. Let's consider, that service times of messages, durations of serviceable work of channels and restoration time of service channels are independent random variables. State of network could be described via vector Z (t) = (z, t) = (d, k, t) = (d(1),d(2),...,d(n),k(1),k(2),...,k(n),t), where d(i) - number of serviceable channels in system S-i, 0 <= d(i) <= m(i), k(i) -messages number in system S-i at the moment t, t is an element of[0,+infinity), m(i) - total number of channels in system S-i, i = (1, n) over bar. By the instrumentality of generating functions approximate expressions for the timedependent state probabilities, average number of messages and serviceable channels are obtained.