Throughout the previous section, we have treated the ``system'' as a single
server: a process enters the system and, within an expected time of 
units, completes and leaves the system. If we view the system as composed of
several servers, rather than just one, we can conduct the same type of analysis,
but now may get different transitions rates. Assume, for instance, that
we have 3 servers in our system; we can picture three identical subsystems,
with a process queue common to all three, from which each subsystem gets its
next process for execution. The state diagram looks just like our previous
one, with one significant difference: the transition rates from state 
to state 
(completion of a process) are three times higher for i>2,
because there are three subsystems operating in parallel. (For i=2, only
two of the subsystems are busy, so the completion rate is 
; for i=1,
it is just 
.) The resulting state diagram (an M/M/3 queueing system)
is shown below.
The analysis of this diagram proceeds exactly as for the diagrams used in the
previous sections, with small modifications made to account for the new
transition rates. (In particular, the utilization ratio of the system is
now 
.)
With this technique we can analyze apparently complex server systems.
Problem: The new Albuquerque airport has a taxi stand of only two
slots. If a taxi arrives and both slots are full, the taxi must depart empty.
Passengers are on a (potentially infinite) queue where the first passenger on
this queue takes the first taxi in the stand, if one is there, or waits for
the next taxi to arrive. Assume that passengers arrive at a rate of 
per minute and that taxis arrive at a rate of per minute ( 
).
(i) What is the probability that a taxi will be turned away because the stand is
full? (ii) What is the probability that a passenger will arrive and find at
least one taxi in the stand?
We can set this problem up as a queue with two identical servers
(the two taxi slots).
In state 
, both taxi slots are taken and no passenger is waiting;
the answer to part (i) is simply 
. In state 
, one taxi slot
is occupied and no passenger is waiting, while in state 
both taxi slots
are empty and no passenger is waiting; finally, in states 
,
both taxi slots are empty and i passengers are waiting.
Thus the answer to part (ii) is simply 
.