Basic Ideas in Queueing Theory

Arrivals, Servers, and Utilization Three concepts underpin every queueing model. The first is Poisson arrivals: when customers arrive independently at a constant average rate $\lambda$, gaps between consecutive arrivals follow an exponential distribution with mean $1/\lambda$, and the count of arrivals in any window of width $t$ follows a Poisson distribution with mean $\lambda t$. These two descriptions are equivalent: if one holds, the other must too. The second key idea is that the exponential distribution is memoryless. Knowing you have already waited $s$ units gives no information about when the next arrival will come. This property makes the math simple, but means that the exponential distribution isn’t a good fit for scenarios where events happen in bursts. Some of the later tutorials will explore models that handle this. The final concept is server utilization. A server completing work at rate $\mu$ has utilization $\rho = \lambda/\mu$, which is the long-run fraction of time…