14 2. SINGLE QUEUES
n D T; (2.1)
where n is the average number of jobs at the queue (waiting and in service), and T is the
average time at the queue.
(b) If we view the waiting space alone as the system, then
L D W ; (2.2)
where L is the average number of jobs waiting for service, and W is the average waiting
(c) If we view the single server alone as the system, then the average number of jobs at the
server is 1 Prob.1 job at the server/ C 0 Prob.no job at the server/ D
where D Prob.1 job at the server/: (2.3)
A server is busy if and only if it is serving a job, so is a measure of server utilization.
Since T D W C
(waiting time plus service time), we have
n D T D
D L C : (2.4)
As is a probability, D
requires that < , i.e., arrival rate is slower than service rate.
is is intuitive; if > , then jobs arrive faster than they can be served, the queue will grow
indeﬁnitely, there is no steady state, and averages have no meaning.
What if D ? e answer depends on whether the inter-arrival and service times are
deterministic. Such details are speciﬁed by the notation we introduce next.
2.2 QUEUE SPECIFICATION
In the Kendall Notation A=S=K=B=N=d ,
A is the arrival process (e.g., M for memoryless, i.e., Poisson arrivals);
S is the service time distribution (e.g., D for deterministic, G for general);
K is the number of servers (possibly inﬁnite);
B is the buﬀer capacity (including jobs in service; by default, B is inﬁnity);
N is the job population size (by default, N is inﬁnity);