Page 107 - DCAP601_SIMULATION_AND_MODELING

Unit 7: Simulation of Queuing System (I)

Poisson arrival pattern: To facilitate the analysis, so assume that the interarrival times of the Notes
customers are such that there is a fixed long-term average time gap between two arrivals. Let
this time be . Let us suppose that the customers arrive independently, and that the probability
of and arrival during any period depends only on the length of that period.
Therefore, the probability of a customer arriving during a very small slice of time h is h/.
Hence the probability of a customer not arriving during time h is (1 – h/). Now let us define
that
f{t) = Probability that the next customer does not arrive during the interval t given that
the previous customer arrived at t = 0, and likewise;
f(t + h) = Probability that the next customer does not arrive during the interval (t + h) given
that the previous customer arrived at t = 0.
Since the arrivals of customers in different periods are independent events (i.e., the queue has no
memory), we can write
 h 
f(t + h) = f(t). 1 – 

  
f(t + h) – f(t) –f(t)
=
h 
Taking limits on both sides as h tends to zero, we get

df(t) –f(1)
=
dt 
The solution of this differential equation is
f(t) = ce  t / ...(1)

Since at time t = 0 an arrival has just taken place, the probability of a nonlinear at t = 0 is 1. That
is, f(0) = 1, and therefore the constant c in Eq. (1) is unity. Thus

f(t) = ce  t / ...(2)
Here is two very simple assumptions, first censtancy of a long-term average and second statistical
independence of arrivals have led to Eq. (2) which gives the probability that the next customer
does not arrive before time t has elapsed since the arrival of the last customer.
The probability that a customer arrives during an infinitesimal interval between t and t + t is
given by the product of the probability that no customer arrives before time t and the probability
that exactly one customer arrives during t. This product is

.
e  t /   t   
  
We can say, the probability density function of the interarrival time is 1
1
.e –1/ ...(3)

The integral

1 t  t /x  t /
 e dt  1 e ...(4)


u
is the probability distribution function.

LOVELY PROFESSIONAL UNIVERSITY 101