Queueing theory uses mathematical models to analyze performance measures of queueing systems. Common measures include average queue length and average delay time. The document provides detailed equations to calculate average queue length for the M/M/1 queueing model. Specifically, it shows that the average number of customers in an M/M/1 queue is equal to ρ/(1-ρ), where ρ is the utilization rate of the server.

1. QUEUEING THEORY
 Queueing Models can provide answers to performance related questions
through mathematical analysis
 Commonly used performance measures
1. Average queue length (average number of customers in system)
2. Average delay time T
1. Compute Average queue length N
 This is the average number of customers in the system.
(Better: This is the average number of customers waiting in the system to get
service....)
Consider the following simple example:
Suppose
50% of the time, the system is empty (0 customer)
50% of the time, the system has 1 customer
Then
the average number of customers in the system is the weighted sum:
N = 0.5 × 0 + 0.5 × 1 = 0.5

2.  In general,
the average queue length
(or)
the average number of customers in system
is equal to:
N = mean (expected) number of customer
= 0 × Ҏ[ 0 customers in system]
+ 1 × Ҏ[ 1 customer in system]
+ 2 × Ҏ[ 2 customers in system]
+ ....
=  
k  k customers in system
0
P
k


(By definition of "expected value")
=
 k (☺Define   P pk k customers in system  )....(1)
0
pk
k



3.  Example:
Compute Average queue length in the M/M/1 queue:
o Recall that the state probability i of the M/M/1 queue is
Ҏ[ k customers in system] = ρk (1 - ρ)
(see: other slide)
 The average (expected) queue length N of the M/M/1 queue is:
N =
 k (☺ Definition of N)& (☺ By (1))
0
pk
k


=
k  
0
(1 ) k
k
 


(By formula, Pk = Ҏ[ k customers in system] = ρk (1 - ρ) )
=
(1 ) k
  k  
.... (2)
0
k


0 1
  1 2 3 4
       
0
know
1
... = (1 )
1
k
k
We
      




Define
1
  
U  
0
(1 ) k
k



Thus

4.  
1 2


        
0
k
k  (1 
) (3)
  
 (  1)(1  )  1  1  ( )  (  1)(1  )  1  1 (  1)  (1  )

2

 
k
dU
d


dU d

d d
  
 
Using (3)
k 2
 
k 1
k 2
 
1

0

0
---------(4)
k
k
   
  

 




   
 k   1

Substitute (4) in (2):
 
N  
   
 
   
   
 
0
1
(1 )
(1 )
(5)
k
k
  

 





       


2
k
1
1
1
Therefore,
the mean number of customers in an M/M/1 queue
is equal to:
ρ
N(M/M/1) = ------- ..... (4)
1 - ρ

5. 2. Average delay time T
 The delay is defined as:
 The average delay time is the average amount of time
that a customer spends in the system.