Queing Model

QUEUING THEORY

HISTORY
• Queuing theory had its beginning in the research work of a
Danish engineer named A.K. Erlang.
• In 1909, Erlang experimented with fluctuating demand in telephonic traffic.

• 8 years later, he published a report addressing the delays in automatic dialing equipment.
• At the end of World War II, Erlang’s early work was extended to more general problems and to business applications of waiting lines.

M/M/1
SINGLE - CHANNEL
WITH POISSON
Azenith Cayetano

THE M/M/1 NOTATION REPRESENTS:
Arrival distribution

M = Poisson

Service time distribution

M = Exponential

No. of service channels open m = 1

QUEUING EQUATIONS:

λ = mean number of arrivals per time period (for example, per hour) μ = mean number of people or items served per time period

SAMPLE PROBLEM 1

Angie is the Branch Manager of Citibank
Lagos and she wants to improve the service of the bank by reducing the average waiting time of the bank’s clients. She was able to determine the average arrival and the average number of clients serviced per hour.

 How many clients are in the bank at any given time? How much time does a client spend in the bank? How many clients are waiting to be served? How much time does a client spend waiting?
 What is the probability that the teller is busy? What is the probability that there are no clients?

DATA TABLE
Given

Description

Value

m

Number of tellers

1

λ

Arrivals per hour

11

μ

Serviced per hour

12

1. Compute the average number of clients in the system (L) at any given time:
L

= λ / (μ - λ)

11

= 11 / (12 – 11)

= 11 clients are in the bank on the average
2. Compute the average number of hours a client spends in the system (W):
W

= 1 / (μ - λ)

1

= 1 / (12 – 11)
= 1 hour is the average time a client is inside the bank

3. Compute the average number of clients waiting in line (Lq):
Lq

= λ2 / μ (μ - λ)

10.0833

= 112 / 12(12