# M/M/C (∞/FIFO) System: Queuing Theory

It is a queuing model where the arrivals follow a Poisson process, service times are exponentially distributed and there are C servers.

Don't get frustrated after seeing these formulas. In the following text, we provide some simple examples that will help you in understanding the model.

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1
---- =
P0
 (λ/μ)n -------- n! + (λ/μ)c -------- c! X 1 ---- 1 - ρ

Where ρ = λ
------

Lq =
 P0 X (λ/μ)c ------- c! X ρ -------- (1 - ρ )2

Wq =
 1 ---- λ X Lq

Ws =
 Wq + 1 ---- μ

Ls =
 Lq + λ ---- μ

#### Example 1

The Silver Spoon Restaurant has only two waiters. Customers arrive according to a Poisson process with a mean rate of 10 per hour. The service for each customer is exponential with mean of 4 minutes. On the basis of this information, find the following:

• The probability of having to wait for service.
• The expected percentage of idle time for each waiter.

#### Solution.

This is an example of M/M/C, where c = 2

λ= 10 per hour or 1/6 per minute.
μ = 1/4 per minute
ρ = 1/3

1
---- =
P0
 (λ/μ)n -------- n! + (λ/μ)c -------- c! X 1 ---- 1 - ρ

1
---- =
P0
 (2/3)n -------- n! + (2/3)2 -------- 2! X 1 ------ 1 - 1/3

1
---- =
P0
 1 + 2 ------ 3 + 1 --- 3

P0 = 1
---
2

The expected percentage of idle time for each waiter.
1 - ρ = 1 - 1/3 = 2/3 = 67%

#### Example 2: M/M/C (∞/FIFO) System

Universal Bank has two tellers working on savings accounts. The first teller handles withdrawals only. The second teller handles deposits only. It has been found that the service times distributions for both deposits and withdrawals are exponential with mean service time 2 minutes per customer. Deposits & withdrawals are found to arrive in a Poisson fashion with mean arrival rate 20 per hour. What would be the effect on the average waiting time for depositors and withdrawers, if each teller could handle both withdrawers & depositors?

Solution.

Given
λ = 20 per hour or 1/3 per minute, μ = 1/2 per minute, c = 2

Case I - Treating depositors and withdrawers as unit of M/M/1 system.

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 Average waiting time of an arrival (Wq) = λ ---------- μ (μ - λ ) Wq = 1/3 ---------- 1/2 (1/2 - 1/3) =   4 minutes

Case II - If each teller handles both depositors and withdrawers.

 P0 = 1/2 Lq = 1/12 Wq = 1 --- λ X Lq Wq = 1/4 minutes

Hence, when both tellers handle both withdrawals & deposits, then expected waiting time is reduced.