Module 8 : Specialized Traffic Studies

Lecture 45 : Queuing Analysis

1

2

3

4

5

6

7

8

Multiple single servers' model

In this model there are N numbers of identical independent parallel servers which receive customers from a same source but in different parallel queues (Compare to M/M/N model. It has only one queue) each one receiving customers at a rate of $\frac{\lambda}{N}$ . Fig. 1 shows how a typical multiple single servers' model looks like.

Figure 1: Multiple single server

Numerical example

Consider the problem 1 as a multiple single server's model with two servers which work independently with each one receiving half the arrival rate that is 150 vehicles/hr.

Solution

Mean arrival rate = $\lambda$ = 150 vehicles/hr. Mean service rate = $\mu=\frac{3600}{10}$ vehicles/hr. Utilization factor = traffic intensity = $\rho=\frac{\lambda}{\mu}=\frac{150}{360}$ = 0.416. The percent of time the toll booth will be idle = P(0) = P(X=0) = $(0.416)^0(1-0.416)=0.584(60 min)$=35.04 min. The average number of vehicles in the system = $E[X]=\frac{\rho}{1-\rho}$= 0.712. The average number of vehicles in the queue = $L_q = \frac{\rho^2}{1-\rho}$= 0.296. The average a vehicle spend in the system = $E[T]= W =\frac{1}{\mu-\lambda}$= 0.0047 hr = 0.285 min = 17.14 sec. The average time a vehicle spends in the queue = $E[T_q] = W_q =\frac{\lambda}{\mu(\mu-\lambda)}$= 0.0022hr = 0.13 min = 8.05 sec

Comparison of the three models

	M/M/1 model	M/M/2 model	Multiple single
			server model
Idle time of toll	8.34	55.2	35.04
booths(minutes)
Number of vehicles	4.98	1.22	0.712
in the system(units)
Number of vehicles	4.01	0.387	0.296
in the queue(units)
Average waiting time	57.6	14	17.14
in system(seconds)
Average waiting time	50	4.64	8.05
in queue(seconds)

From the Table 1 by providing 2 servers the queue length reduced from 4.01 to 0.387 and the average waiting time of the vehicles came down from 50 sec to 4.64 sec, but at the expense of having either one or both of the toll booths idle 92% of the time as compared to 13.9% of the time for the single-server situation. Thus there exists a trade-off between the customers' convenience and the cost of running the system.