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  "body": "\n## Definition\n\nLet $A$ and $B$ be [[probability distributions|0DVDCMA2DK]] on $(0, \\infty)$ and let $c \\ge 1$ be an integer.\nA *single-station queue* $A/B/c$, in Kendall's notation, is the following model:\n\n1. customers arrive at times $0 \u003c \\tau_1 \u003c \\tau_2 \u003c \\cdots$ whose gaps $\\tau_{n+1} - \\tau_n$ and first time $\\tau_1$ are [[independent and identically distributed|00MRN0FARV]] with law $A$;\n2. customer $n$ needs a service time $\\sigma_n$, where $\\sigma_1, \\sigma_2, \\ldots$ are independent with law $B$ and independent of the arrivals;\n3. there are $c$ servers and an unlimited waiting room, and customers are served in order of arrival by the first free server.\n\nThe queue length $Q(t)$ is the number of customers in the system at time $t$, waiting or in service.\nQueueing theory studies such models and their networks.\n\n## Remarks\n\nThe letter $M$ stands for an exponential law, which makes the model Markov, and $D$ for a deterministic law.\nWith mean gap $1/\\lambda$ and mean service time $1/\\mu$, the traffic intensity is $\\rho = \\lambda / (c \\mu)$.\nExample: the [[M/M/1 queue|GTFRPF5J2K]] takes $A$ exponential of rate $\\lambda$, $B$ exponential of rate $\\mu$ and $c = 1$; then $Q$ is a birth–death process.\nNon-example: if customers who find the server busy leave at once, the model is a loss system, not an $A/B/c$ queue, since clause 3 asks for an unlimited waiting room.\nFor the $A/B/1$ queue with finite means, $\\rho \u003c 1$ ensures that the waiting times converge in law to a finite limit, and $\\rho \u003e 1$ makes them tend to infinity.\n",
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