Investigation of HM-network with stochastic incomesfrom transitions between their states

HM-network, i.e. Markov queueing network of arbitrary topology with incomes, which consistsof n multi-channel systems is observed. Poisson flow of rate λ enters the network. The conditionalservice rates of jobs in network systems ( ( )) i i k t ƒ 1, i n = , depend on number of jobs inthem. Job during it transition from one queueing system to another brings some stochastic incomewith given mean value to the last one and the income of the first system decreases by this value.Income of system Si at the moment t+ ¦t can be found from relation( ) () (, ) i i i V t t V t V t t + ƒ = + ƒ ƒ ,where ¦Vi(t,¦t) - income change of system Si at the interval [t, t+¦t). For finding of this valueprobabilities of all events which can appear during time t ƒ and income changes of system Siwhich connect with these events were derived. Under the assumption that min( ( ), ) i i M kt m =min( ( ), ) i i N t m = , where ( ) i N t - average number of jobs (waiting and serving) in system i S atthe moment t , 1, i n = , approximate relation for change of mean income of system Si. and systemof heterogeneous linear ODE with discontinuous right parts for ( ) i N t , 1, i n = , were obtained.01( ) min( ( ), ) min( ( ), )nij ji j j i i i ijj idN t p N tm Ntm pdt == ƒ −ƒ + ƒ , 1, i n = .For expected incomes of network systems heterogeneous linear ODE of the first order werereceiveddv t N t m p b p b N t m p a p a cdt = =⎛ ⎞= −ƒ + + ƒ + ƒ + ⎜ ⎟ ⎜ ⎟⎝ ⎠ .In numerical example expected incomes for queueing network with central system were obtained.

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