# M/M/1 (N/FIFO) System : Queuing Models

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.

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 ρ = λ/μ Po = 1 − ρ -------- 1 − ρN + 1 Ls = ρ -------- 1 - ρ − (N + 1)ρN+1 ----------- 1 − ρN + 1 Lq = Ls - λ/μ Wq = Lq ---- λ Ws = Ls ---- λ

## Example: M/M/1 (N/FIFO) System

Students arrive at the head office of Universal Teacher Publications 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.

 λ = 30 --------- 60 X 24 = 1/48 students per minute μ = 1/36 students per minute ρ = 36/48 = 0.75 N = 9 Po = 1- 0.75 ------------- 1- (0.75)9 + 1 = 0.26 Ls = 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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