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M/M/2 Queue

Unlike the M/M/l queue where there was only one server, in this variation, there are two servers serving in the queue.

Probability of Having n Customers in the Queue As before, letting Pn (t) represent the probability that there are n people in the queue at time t (including the person being served) where n= 1, 2, 3, ...,equations (9.3a) and (9.3b) become

(9.5a)

(9.5b)

(9.5c)

(9.5d)

Taking limits as , one can arrive at the following analogs for equations (9.4a) and (9.4b):

(9.6a)

(9.6b)

(9.6c)

For the queue to reach stationarity conditions (i.e., be time independent and stabilize), one has that (for n = 0,1,2,3,...). Doing this allows one to use equations (9.6a) to (9.6c) to obtain for n = 1, 2, 3, ..., where . Using the identity , it follows that

Expected Waiting Tine To find the expression for the expected waiting time (i.e., the time taken for an arriving customer to get his service completed), one needs a waiting-time distribution. As before, letting W represent the waiting time and f(w) the pdf of W (where w > 0), it readily follows that

From the above, it follows that the expected waiting time for such a queue is given by the expression, which simplifies to

Little's Formula to Compute Expected Waiting Time As can be seen from the discussion presented thus far, in increasing the number of servers from one to two, the problem associated with finding the expected waiting time quickly gets more complicated. Despite the complexity associated with the derivation of f(w), one can extend the problem to entertain more servers and calculate the expected waiting time until the completion of service using Little's formula (1961). Little's formula simply states that

(9.7a)

where

E (waiting time) refers to the expected time taken for a newly arriving customer get his service completed.

E (number in system) refers to the expected number of customers in the queuing system.

A related variation of the formula that is given in equation (9.7a) is E (number in queue) = λ * E(waiting time until start of service) (9.7b)

where

E (waiting time until start of service) refers to the expected time taken for a newly arriving customer to get his service started.

E (number in queue) refers to the expected number of customers in the queue waiting to get their service started.

Another formula that is useful in connecting equation (9.7a) with equation (9.7b) is

(9.7c)

To apply equation (9.7a), consider the M/M/l queue. For this type of queue, it readily follows that

Hence the E (waiting time) == (which agrees with the result obtained for the M/M/l queue).

To apply equations (9.7b) and (9.7c) to the M/M/2 queue, first consider the size of the queue that is waiting to be served. Then it is easy to see that

Hence, using equation (9.7c), one gets E (waiting time) (which again agrees with the result obtained for the M/M/2 queue).

Percentage Of Downtime To calculate this quantity, it is important to first observe that both the servers would have nothing to do if no one is in the queue. Thus, the percentage of downtime is simply given by the expression . Furthermore, the percentage of time that both the servers would both be busy is given by the expression

 
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