On mixed time-changed Erlang queue
Pub. online: 9 June 2026
Type: Research Article
Open Access
Open Access
Received
21 August 2025
21 August 2025
Revised
27 March 2026
27 March 2026
Accepted
23 May 2026
23 May 2026
Published
9 June 2026
9 June 2026
Abstract
We study a time-changed variant of the Erlang queue by taking the first hitting time of a mixed stable subordinator as the time-changing component. We call it the mixed time-changed Erlang queue. We derive the system of fractional differential equations that governs its state probabilities. The explicit expressions for the state probabilities of mixed time-changed Erlang queue and their Laplace transform are derived. Also, an equivalent representation of this time-changed queue in terms of phases is provided, and its mean queue length is obtained. Some of its distributional properties such as the distribution of its inter-arrival times, inter-phase times, service times and busy period are derived. Later, its conditional waiting time is discussed and some plots of sample paths simulation are presented.
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