Page 119 - DCAP601_SIMULATION_AND_MODELING

Unit 7: Simulation of Queuing System (I)

Notes

Task Differentiate between waiting time and idle time of the server.

Single-server M/M/1 Model

Construction of Model

In the minitype super-market, there is a cash desk, and customers reaches the desk in random.
Provided that the customer reaches as the cashier is idle, the customer will pay off immediately
and leave. If the cashier is busy as the customer reaches, the customer will have to wait in the
line, nay, no person leaves without waiting. Once the customer enters in the queue, he will
receive service according to FCFS rule. The customer departs after receiving once service. The
interval of customers arriving desk obeys negative index distribution with average value equaling
to 5, and service time of each customer complies with normal distribution with average value
being 1.6 and standard deviation being 0.6. Time calculates at minute, and service time must be
positive.
Figure 7.7: Single-server M/M/1 Model

Waiting Priority Service Serviced Customers
Population Customers Line Facility rule

Simulation of Model

1. The creation of random number. It is desired to describe random factors in the objective
process in nearly all of the simulation process like arrival process and service process in
actual system. Random number comes from collectivity in random. In this model, there
are the interval of customers’ arrival and the service time of each customer. The former
obeys negative index distribution with average number being 2.5, and the latter obeys
normal distribution with average value being 1.6 and standard deviation being 0.6. The
symmetrical-distribution random number U(0,1) must create ahead of the creation of
specific-distribution random number.

(a) Creating the algorithm of obeying negative index distribution with Inversion of
Transforms method the density function of negative index distribution:
–x
f(x) = e , x0 × E(X) = 1/
Its distribution
Function:
x
–
t
F(X)  e dt  1 – e – x, x  0,i.e.,
 0
R = F(X) = 1 – e –x
Through inversion transform, we can obtain:
X= –1/ ln(1–R) Let u=1–R, thus u is a random number in (0,1)
(R obeys symmetrical distribution in (0,1))

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