Page 110 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes Clearly, at any given time t the probability of the service counter being busy is

average service time 
= =  ... (7)
average arrival gap 

 is an important ratio. It is called 'the utilization factor of the service facility.
It immediately follows from Eq. (7) that the probability of finding the service counter free is
1 –  ...(8)
Let P (t) be the probability of exactly n customers being in the system at time I. Let h > 0 be a very
n
small slice of time. The probability of one customer arriving and no customer departing during
interval h is

h  h 
. 1 

   
Likewise, the probability of one customer arriving and one customer departing during interval
h is

h h
.
 
The probability of no customer arriving and one customer departing is

 h  h
 1   .
   
and finally of no customer arriving and no customer departing is

 h  h 

 1   . 1  

     
Since h is so small that no more than one departure and no more than one arrival can take place,
these are the only possible changes that could occur during an interval h.
For the queueing system to have n customers at time (t + h), it must have had either n or (n + 1)
or (n – 1) customers at time t. The probability that there are n customers in the system at time
(t + h) can therefore be expressed as the sum of these three probabilities. Thus for any n > 0 we
can write

 h  h  h h  h  h h  h


P (t h) P (t). 1   . 1    P (t). .  P n 1 (1) 1   .  P n 1 (t). . 1  





n
n


n
               
From this we get
P (t h) P (t)  1 P (t)  1 P (t)–  1  1 P (t)  h [2P (t) P (t) P (t)]


n
n






h  n 1  n 1       n  n n 1 n 1
Taking the limits of both sides of this equation when h tends to zero


P (t h) P (t) 1  1 1  1
lim n n  P (t)     P (t)  P


h 0 h  n 1     n  n 1
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