Queuing Model [M/M/1:K,FIFO] Single Server and finite queuing Capacity with Solved Problems

[M/M/1: K, FIFO] is one of the Queuing Model with Single Server and finite(limited) Capacity. Only K customers can be accommodated in the system. In this model, customers arrive in Poisson fashion and the service follows exponential fashion.

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Important Formulae of [M/M/1: K, FIFO]: (Operating Characteristics)

 Read also:

Little's Formula

1. Traffic Intensity:

It is the ratio of the average arrival rate to the average departure rate. It is also the ratio of mean service time to mean arrival time. It is also called the Utilization Factor. It gives the proportion of the time server is busy. It is denoted by rho(ρ).

2. Effective arrival rate:

It is denoted by λ_e .
 λ_e  = λ(1 - P_k) = μ(1 - Po)

3. Steady State Probability (Limiting probability): 

It is also the probability of Queue being in a state 'n'. It is denoted by 'Pn'.

4. The probability that the Server is idle (no arriving customers):

 It is denoted by 'Po'.

5. The average number of Customer in the System: 

It is denoted by Ls. 

6. The average number of Customer Waiting in the Queue.

7. The amount of Time Customer Spend in the System

8. The Amount of Time Customer spend waiting in Queue

9. The probability that arriving customer get no waiting seat(Turn away) = P(N>=K) = P_k 

10. The probability that arriving customer will get a waiting seat = 1 - P_k

11. The Probability that an arriving customer will not have to wait (no customer) = Po

12. The Probability that an arriving customer will have to wait  = 1 - Po

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Solved Problems of (M/M/1: K, FIFO)

Solved Problem 1: During 8:00 to 10:00 AM each morning, a train arrives every 20 minutes in a yard. The service times have an average of 36 minutes for each train. If the capacity of the yard is 4 trains only. Find:

i. The probability that the yard will be empty. 

ii. The average queue size. 

= Solution:

Here Given, 

System Capacity(K) = 4

arrival rate (λ) = 1/20 per minute

Service rate (μ) = 1/36 per minute

So, Traffic intensity (ρ) = λ/μ

                                    = 1.8

Model Identification:

This is [M/M/1: K, FIFO] queuing model. Since there is a single server and capacity of the system is finite(limited).

i. The probability that the yard will be empty = Po
Po = (1-ρ)/(1-ρ^K+1) 
        = 0.044
ii. The average queue size = Lq = Ls - λ_e/μ (Use above formula for Ls and λ_e )

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Solved Problem 2: Customer arrives at a drive-in-bank in Poisson fashion at the rate of 10 per hour. and the service rate is 12 per hour. The space in front can accommodate a maximum of 3 cars and others can wait outside this space.
i. Find the probability that arriving customer can drive directly to space in front of drive-in-bank?
ii. Find the probability that arriving customer will have to wait outside the indicated space?

= Solution:
Here Given,

System Capacity(K) = 3

arrival rate (λ) = 10 per hour
Service rate (μ) = 12 per hour

So, Traffic intensity (ρ) = λ/μ

                                    = 0.83
i. the probability that arriving customer can drive directly to space in front of drive-in-bank (server is idle) = (1-0.83)/(1-0.83³⁺¹) = 0.32

ii. the probability that arriving customer will have to wait outside the indicated space(turn away) = P(N>=K) = P_k = P₃  = 0.1863 (using formula)

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So these are a complete theory and Queuing Theory Solved problems related to the Queuing Model [M/M/1: K, FIFO].

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