A rate balance principle and its application to queueing models
We introduce a rate balance principle for general (not necessarily Markovian) stochastic processes. Special attention is given to processes with birth-and-death-like transitions, for which it is shown that for any state n, the rate of two consecutive transitions from \(n-1\) to \(n+1\) coincides with the corresponding rate from \(n+1\) to \(n-1\). We demonstrate how useful this observation is by deriving well-known, as well as new, results for non-memoryless queues with state-dependent arrival and service processes. We also use the rate balance principle to derive new results for a state-dependent queue with batch arrivals, which is a model with non-birth-and-death-like transitions.
KeywordsRate balance G/M/1 M/G/1 Birth–death process Batch arrivals Conditional distribution Residual lifetime
Mathematics Subject Classification60G17 60K25
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