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Waiting Line Models

The M/M/1 (N/FIFO) system

It is a queuing model where the arrivals follow a Poisson process, service times are exponentially distributed and there is only one server. Capacity of the system is limited to N with first in first out mode.

The first M in the notation stands for Poisson input, second M for Poisson output, 1 for the number of servers and N for capacity of the system.

r = l/m

P_o =	1-r -------- 1-r^{N + 1}

L_s =	r -------- 1 - r	–	(N + 1)r^N+1 ----------- 1-r^{N + 1}

L_q =	L_s - l/m

W_q =	L_q ---- l

W_s =	L_s ---- l

Example

Students arrive at the head office of www.universalteacher.com according to a Poisson input process with a mean rate of 30 per day. The time required to serve a student has an exponential distribution with a mean of 36 minutes. Assume that the students are served by a single individual, and queue capacity is 9. On the basis of this information, find the following:

The probability of zero unit in the queue.
The average line length.

Solution.

l =	30 --------- 60 X 24
= 1/48 students per minute

m = 1/36 students per minute

r = 36/48 = 0.75 N = 9
P_o =	1- 0.75 ------------- 1- (0.75)^{9 + 1}
= 0.26

L_s =	0.75 -------- 1 - 0.75	-	(9 + 1)(0.75)⁹⁺¹ ---------------------- 1- (0.75)^{9 + 1}
= 2.40 or 2 students.

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