Investigation of queueing network with unreliable systems at transient regime

The open exponentional queuing network with unreliable systems which functioning undercondition of heavy loading is investigated. The Poisson flow of rate ƒ(t) enters the network. Theservice time of messages in each of network systems has exponential distribution with parameterƒi (t ) , i= 1,n. Service channels are exposed to random failure and serviceable work time of eachchannel of system Si has exponential distribution with parameter ƒi (t ) , i= 1,n. After failure theservice channel immediately starts to be restored and restoration time also has exponential distributionwith parameter ƒi (t ) , i= 1,n. Let's consider, that service times of messages, durations ofserviceable work of channels and restoration time of service channels are independent randomvariables.State of network could be described via vectorZ (t)=(z,t)=(d,k,t)=(d1,d2,…,dn,k1,k2,…,kn,t),where di − number of serviceable channels in system Si , 0≤di≤mi, ki − messages number insystem Si at the moment t , t[0,+) , mi − total number of channels in system Si , i= 1,n.By the instrumentality of generating functions approximate expressions for the timedependentstate probabilities, average number of messages and serviceable channels are obtained.

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