Queuing Systems - Yousef's Notes

#Labeling Convention (Kendall-Lee)

State of the system: number of customers in the queueing system (includes customers in service)
Queue Length: number of customers waiting for service = state of the system - number of customers being served.
s = number of servers in the system.
N(t) = state of the system at time t.
$P_n(t)$: probability that exactly $n$ customers are in the queuing system at time $t$.
L = expected number of customers in the system.
$L_q$ = expected number of customers in the queue.
$\lambda$ = mean arrival rate (expected number of arrivals per unit time)
$\mu$ = mean service rate for a busy server.
$1/\lambda =$ expected inter-arrival time
$1/\mu =$ expected service time
$\rho = \frac{\lambda}{s \mu}$ is the utilization factor for the service facility (expected fraction of time the system’s service capacity is being utilized by arriving customers) Assuming that $\lambda \text{ and } \mu$ are independent of the state of the system.