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.
| ρ = λ/μ | |||
| Po = | 1 − ρ -------- 1 − ρN + 1 |
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| Ls = | ρ |
− | (N + 1)ρN+1 ----------- 1 − ρN + 1 |
| Lq = | Ls - λ/μ | ||
| Wq = | Lq ---- λ |
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| Ws = | Ls ---- λ |
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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 |
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| = 1/48 students per minute | |||
| μ = 1/36 students per minute | |||
| ρ
= 36/48 = 0.75 N = 9 |
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| Po = | 1- 0.75 ------------- 1- (0.75)9 + 1 |
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| = 0.26 | |||
| Ls = | 0.75 |
- | (9 + 1)(0.75)9+1 ---------------------- 1- (0.75)9 + 1 |
| = 2.40 or 2 students. | |||
