Question

7. The Canara Bank drive-thru teller window can serve a customer at an average of 4 minutes per customer. Service time has a negative exponential distribution. Customers arrive in their cars at a rate (Poisson distributed) of 12 per hour and form a single waiting line: a. Determine the average waiting time, the average queue length, and the probability that there is no customer in the system. b. If Canara Bank decides to open a second drive-thru teller window with the same service rate as the first one, how will your answers to Part a change?

Answer 1

Here number of server (c) = 1

Service rate = 4 min/customer

Customer arrival rate = 60/12 = 5 min/customer

$\lambda$ = 1/ customer arrival rate = 1/5 = 0.2

$\mu$ = 1/service rate = 1/4 = 0.25

= 0.2/0.25 = 0.8

ρ= λ/μ

a) Using formulas from queing theory

Average queue length ( Lq )=
p/(1-P)
= = 3.2

Average waiting time ( Wq ) = Lq /$\lambda$ = 3.2 / 0.2 = 16 min

Probability of having 0 customer in the system = 1- $\rho$ = 1-0.8 = 0.2

2) For c=2,

$\rho= \lambda /c\mu$ = 0.8/2 = 0.4

Let's find $P_{0}$ using following formula:

e 号(s): Eco Cas)

Hence = $P_{0}$ = 0.49

Average queue length ( Lq ) = $\frac{P_{0}(\frac{\lambda }{\rho })^{c}\rho }{!c(1-\rho)^{2}}$ = 0.174

Average waiting time = $L_{q}/\lambda$ = 0.87