Queing Model

Topics: Automated teller machine, Opportunity cost, Queueing theory Pages: 23 (1984 words) Published: April 27, 2014
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 – 11)
= 10.08 clients are waiting in line on the average

COMPUTATION:
4. Compute the average number of hours a client waits in line (Wq): Wq

= λ / μ (μ - λ)

0.9167

= 11 / 12(12 – 11)

= 0.92 hour is the average time a client is in line
5. Compute the probability that the teller is busy (ρ):
(ρ)

=λ/μ

0.9167

= 11 / 12
= 92% is the probability that the teller is busy

6. Compute the probability that there are no clients in the bank (Po): Po

= 1 - (λ / μ)

0.0833

= 1 – (11 / 12)
= 8% is the probability that the teller is idle

STATISTICS TABLE
Equations

Description

Value

L

Numbers of clients in the system

11

W

Average hours client spends in the system

1

Lq
Wq

Average clients waiting in line
Average waiting hours per client

10.0833
0.9167

(ρ)

Probability that the teller is busy

0.9167

Po

Probability that there are no clients

0.0833

SAMPLE PROBLEM 2

Angie, the Branch Manager of Citibank
Lagos in Example 1, knows that there is
also an opportunity cost for clients who
are idle while waiting in line. She was
able to determine the teller’s labor
cost as well as the average
opportunity cost of clients who are
waiting.

 How much is the total cost per
shift?

DATA TABLE
Given

Value

Cs

Service cost per teller ($/hour)

5

Cw

Cost of waiting ($/hour)

6

h

COMPUTATION:

Description

Working hours per shift

8

1. Compute the total service cost (Ts):
Ts / hour
= mCs
= 1(5)
Ts / shift
= Ts / hour(h)
= 5(8)

=5
= 40

2. Compute for the total waiting cost (Tw):
Tw / hour
= λWqCw
= 11(0.9167)(6)
Tw / shift
= Tw / hour(h)
= 60.50 (8)

= 60.50
= 484.00

3. Compute for the total cost (Tc):
Tc / hour
Tc / shift

= Ts / hour + Tw / hour
= Ts / shift + Tw / shift

= 5.00 + 60.50
= 40.00 + 484.00

= 65.50
= 524.00

COST TABLE
Cost

Per Hour

Per Shift

Total service cost ($)

5.00

40.00

Total...
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