Two-Dimensional Markov Processes and Their Applications to Memoryless Queues

We have already seen Markov processes in which each state is represented by a vector. These Markov processes offer a convenient way to refer to a state (in fact, this was nothing more than giving names to states). Such was the case, for example, when we dealt with open and closed networks of queues, and with multiclass single-server queues. In this respect there is nothing new in this chapter where the states will be represented by two-dimensional vectors; that is, a state will be referred to by the pair of nonnegative integers (i, j). However, in an attempt to generalize some results from one-dimensional birth-and-death precesses, we will impose the following restrictions on the model:

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