QUeuing model

1. Model II : single channel, Poisson Arrival with Exponential
service, finite Population model (limited Sources Model)
(M/M/1): (FCFS/n/N)
N= Limited Number of customer in the system
1. Probability that server is idle ( or empty system)
2. Probability that there are n customer in the system
Pn= ρN P0 where n= 0,1………N

2. Model II : single channel, Poisson Arrival with Exponential
service, finite Population model (limited Sources Model)
(M/M/1): (FCFS/n/N)
3. Average no. of customer (expected ) in the system
4. Average no. of customer in the queue
Lq= Ls- ( λe /μ)
Where ,Traffic intensity depends on mean arrival entry in system (ie. Effective
arrival rate) λe= λ(1-Pn)

3. Model II : single channel, poisson Arrival with Exponential
service, finite Population model (limited Sources Model)
(M/M/1): (FCFS/n/N)
5. Average waiting time of customer in system
6. Average waiting time of customer in the queue

4. Model II Formulae (M/M/1): (FCFS/n/N)

5. Example :
(M/M/1):(N/FCFS)
Example 1: Trains arrive at the yard every 15 minutes and service time is 32
minutes . If the capacity of yard is limited to 4 trains . Find
- Probability that yard is empty
- Average number of trains in the system

6. Solution:
• N=Capacity of yard= 4 trains
• Arrival rate=
λ = 1/ 15 trains / Minute
= (1/15)*60 trains/hr
= 4 trains/hrs
• Service rate= μ = 1/32 trains/ minute
= (1/32)*60 train/ hrs
= 1.88 trains/hrs

7. - Probability that yard is empty
ρ = 4/1.88= 2.1
P0= (1-2.1)/ 1-(2.1)4+1
=
( -1.1)/1- 40.84
= -1.1/-39.84
= 0.027 i.e 2.7%
- Average number of trains in the system

8. - Average number of trains in the system

9. Example 2: Patient arrive at a clinic according to a poisson distribution at rate of 30
patient per hour . The waiting room does not accommodate more than 14 patients
. Examination time per patient is exponential with mean rate 20 per hour
1. Find effective arrival rate at the clinic
2. What is the expected waiting time until a patient is discharged from
Solution:
N= 14 patients
• Arrival rate= λ = 30 patient /hrs
• Service rate = μ = 20 patient / hrs
• ρ = λ / μ = 30/20= 1.5

10. 1. effective arrival rate at the clinic =
λe= λ (1 –Pn) -----------(i)
Where
Pn = ρN * P0
Put in eq(i)
λe= λ (1 –Pn) = 30 ( 1- 0.33)= 30 x( 0.67)= 20. 1 ie. 20 patients

11. 2. What is the expected waiting time until a patient is discharged from
Put Ls value in (ii)
Ws= Ls/ λe = 12.03/ 20.1 = 0.598 hrs
-------- eq( ii )