For all of the previous distributions, the average time spent in the system (T) consists of an average service time Ts, plus a waiting time (due to contention) Tw:
T = Ts+Tw.
If N is the average queue length (including request being serviced) and T is the average time (including service time) for completion of service:
N = lT (Little's result).
Since N consists of items in the queue and an item in service (Figure 6.17), then
N = Q + r.
Recall that the server is occupied r part of the time. Hence the average number of items in service is r. Since
N = lT,
then (see also Figure 6.17)
Q + r
=
l(Ts + Tw),
Q + r
=
l(1/m + Tw),
Q + r
=
r + lTw,
Q
=
lTw.
Note that Q = N - r or N = Q + r.
Queues that result from the buffering of requests that do not affect the request rate l are called open queues,Qo. Again, for these queues:
