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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

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