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Some inequalities for a multiphase system. (Russian) Zbl 0544.90026
The paper contains some estimates of a capacity of multiphase (tandem) queueing systems having limited intermediate queues. A demand served in a previous phase cannot be set free if there are no places in the following queue (a blocking effect). It is proved that a value T exists such that if the intensity of the demand flow is less than \(T^{-1}\), a stationary regime is achieved and it is not achieved if the intensity is larger than \(T^{-1}\). By definition, the value \(T^{-1}\) is the system capacity and the value T is the mean effective service time. It is stated that T is a monotone non-decreasing function of the service time in each phase and the quantities of the places between the phases (the time \(\xi_ 2\) is not less than the time \(\xi_ 1\) if \[ \int^{\infty}_{x}(1-F_ 1(t))dt\leq \int^{\infty}_{x}(1-F_ 2(t))dt\quad for\quad all\quad x \] where \(F_ 1\), \(F_ 2\) are the distribution functions of \(\xi_ 1\), \(\xi_ 2)\). The main result gives an upper bound for T in a system having 3 phases, k places before the 2-nd phase and r places before the third one. The bound coincides with the correct value of T if \(r=0\) and the distribution of service times is exponential.
Reviewer: A.A.Pervozvanskij

90B22 Queues and service in operations research
60J25 Continuous-time Markov processes on general state spaces