The M(m)/M/K and M(m)/M/ models with synchronous fluctuation of traffic intensity are considered. The phase process is assumed to make changes according to an irreducible m-phase Markov chain. In contrast to the model with asynchronous fluctuation of parameters, phase changes may occur in synchronization with an arrival or beginning of a customer's service. We study mainly the steady-state regime of our models, and observe that, in general, closed form solutions for the limiting probabilities are difficult to obtain but their numerical computation is rather straightforward. We give a necessary and sufficient condition for the steady-state to be attained. For the model M(m)/M/K, it is shown that, for the case where the traffic intensity of one phase is greater than one (even if the average traffic intensity is less than one) the average queue length approaches infinity as the fluctuations among phases gets more sluggish. However, for the case where the traffic intensity for all phases is less than one, the queue length is moderate and not dependent as much on the rate of fluctuation among phases. Numerical examples are given and discussed. Finally, we point out that, our models may be more tractable than the asynchronous ones, when we try to generalize them to the case of general inter-arrival, service, or sojourn time distribution.

Multi-server queueing systems with traffic intensity which vary according to an irreducible Markov chain are considered. We will show that the probability distribution for the number of customers in these systems can be expressed by an algebraic sum of geometric series with appropriate coefficients satisfying some interesting properties. To study these properties a detailed analysis of the denominator of the partial generating functions of the number of customers in the system is presented. These coefficients allow us to explain the behavior of the system under different traffic conditions. First, we derive a general expression for the probability distribution and then compare this result with that of the Zukerman and Rubin's model. One special section is devoted to the case of an arbitrary number of phases. Some numerical results are also provided and discussed to support the theoretical results.

A study is presented on the generalization of the M/M/1 model with synchronous fluctuation of traffic intensity for the case of more than one server. The parameters of the queueing system fluctuates between two phases and the phase process is assumed to make changes according to an irreducible Markov chain. Phase changes may occur only when a customer arrives to the queue, further, his service time is also determined at his arrival instant. We compare our results to that of the asynchronous one and the comparison clearly shows that we can approximate the performance of the asynchronous model by that of the synchronous one and vice-versa. However, we point out that, to extend the M/M/K (M/M/) model with fluctuating parameters to the case of general interarrival, service, or sojourn time distribution, is easier in the case of the synchronous model than in the case of the asynchronous one since the latter introduces very considerable analytic complications. Two models, the M/M/K and M/M/, with synchronous fluctuation of parameters are analyzed in depth. Explicit results for the expected queue length are presented and discussed. Finally, numerical examples are used to support the theoretical analysis for a variety of traffic conditions.