On queueing systems with failures and several types of input flows (Q1969169)

The authors describe a servicing process of various call flows by means of queueing theory. They consider a queueing system with \(m\) servers. In this system \(m\) \((m\leq n)\) input flows of call arrive simultaneously. Arrival moments of \(i\)th flow form Poisson flow of rate \(\lambda_{i}\), \(i=1,2,\ldots,m.\) At the moment of arrival \(n_{i}\) calls arrive into the system. If at least \(n-n_{i}+1\) servers are occupied, then the calls get lost, otherwise they occupy \(n_{i}\) arbitrary vacant servers. Two variants of call service are considered. \(S\langle m,n\rangle\) variant is one: if call batch of size \(n_{i}\) corresponding to the \(i\)th input flow has arrived into queueing systems, then service times of these calls are independent and exponentially distributed with parameter \(\mu.\) \(R\langle m,n\rangle\) variant is one: if at arrival moment of the call batch corresponding to the \(i\)th input flow \(n_{i}\) vacant servers can be found, then calls of this batch occupy \(n_{i}\) servers at time distributed by an exponential law with parameter \(\mu_{i}.\) The stationary regime of service process for \(S\langle m,n\rangle\) and \(R\langle m,n\rangle\) queueing systems is investigated. A calculation scheme of failure probabilities for calls of each flow is constructed. For example the following theorem is proved: The stationary probabilities \(\pi_{k}\), \(k=0,1,\ldots,n\), of the \(S\langle m,n\rangle\) system have the form \(\pi_{k}=\alpha_{k}\{\sum_{i=0}^n \alpha_{i}\}^{-1}\), where \(\alpha_{k}\), \(k=0,1,\ldots,n\), are determined by the system of recursion relations \[ k\alpha_{k}=[(k-1)+\Delta_{k-1}]\alpha_{k-1}- \sum\limits_{i=1}^m \rho_{i}\alpha_{k-1-i}, \quad k=0,1,\ldots,n, \] with initial conditions \(\alpha_{-m} =\ldots= \alpha_{-1}\), \(\alpha_0=1\), \(\Delta_{i}\) and \(\rho_{i}\) are known parameters depending on \(\lambda_{i}\).