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It is known that the multiserver of systems with losses and any holding time M/G/s (0) is equivalent to Markovsky model M/M/s (0), and the probability of blocking here is set by the formula of Erlanga of V.Krome togo, probability of blocking for system with the limited number of inputs from n sources M (n)/G/s (0) set by losses under the formula of Engseta. The given properties concern to робастности service time.
It is known that system M/G/1 has a steady condition, if and only if offered transport loading a = r = lh <1 erl where h there is an average holding time, and r - system operating ratio. It can be understood intuitively because the server can serve 1 erl, as a maximum.
Let's choose some call and we will mark it as a test call (fig. 10 see). Probability of that the server is occupied, when the test call will arrive, on property of 3 loadings and PASTA(Poisson arrivals see time average,т.е это Пуассоновское поступление вызовов, наблюдаемое за среднее время) is equal and. Time until will come to the end call service, is residual service time. From here, having designated an average residual holding time a symbol, average of expecting calls - and an average waiting time - we have the following dependence at an order of service of calls FIFO: formula (18)
On the right side of expression (18) the first product corresponds to average time for a call in service if some has to be finished, and the second product corresponds to service of those expecting calls which stand in a queue ahead of a test call.
Using the formula of Littla = l and solving dependence (18), we will receive an average waiting time formula (19)
From expression (16) we will receive average residual time formula (20)
Where Cs2 = σs2/h2 - is SCV service time (service); σs2 - Is a dispersion of service time (service).
Substituting average residual time (20) in an average waiting time (19), we will receive the formula of Poljacheka-Hinchina: formula (21)
In stochastic process time moment in which property марковости keeps, called as a pulsing point. For system M/G/1 the epoch of deportation (clearing) in which the call comes to the end and leaves system, becomes a pulsing point.
Markovsky process with discrete space of conditions is called as the Markovsky chain. In the Markovsky chain all times of epoch in which a condition changes, become pulsing points. On the other hand, stochastic process is called as the introduced (enclosed) Markovsky chain if pulsing points take root or put during special time of epoch, such as deportation (clearing) of a call in systems of type M/G/1.
In the introduced Markovsky chain (fig. 11 see) a condition of probability Пj* just after clearing (call deportation) to equally condition of probability Пj just before call receipt in a steady condition. From PASTA(Poisson arrivals see time average,т.е это Пуассоновское поступление вызовов, наблюдаемое за среднее время) follows that if in system M/G/1 exists j calls, then formula
Distribution of waiting time W (t) in system M/G/1 define under the formula of Benesha. Can be shown that the equation of integral of Volterra satisfies to distribution W (t). For system M/D/1 analytical expression for function of distribution of waiting time W (t) is received, and for system M/G/1 (m) the formula of calculation of an average waiting time is received.
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