1 Queuing Analysis Overview What is queuing analysis? - to study how people behave in waiting in line so that we could provide a solution with minimizing.

1

Queuing AnalysisOverview

• What is queuing analysis?- to study how people behave in waiting in line so that we

could provide a solution with minimizing waiting time and resources allocation

Two elements involved in waiting lines (to p2)

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Two elements involved in waiting lines

1. Arrival rate, • Rate of people joining to the queue

2. Service rate, • Rate of service that service provided

How do they applied in a real life? (to p3)

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Queuing Analysis

(p9)

Service rate,

Arrival rate,

This phenomenon is known as Single-server Waiting Line

How to study it? (to p4)

HKBU

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The Single-Server Waiting Line System

The Single-Server Model

We assumed that

1. An infinite calling population (that is many people can join the queue)

2. A first-come, first-served queue discipline

3. Poisson arrival rate

4. Exponential service times

- Sympology:

= the arrival rate (average number of arrivals per time period)

= the service rate (average number served per time period)

Again, we assumed that (< ) or we can never finish service customers before the end of the day! (to p5)

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The relationship between and • We adopted a “birth-and-death” process to

study their relationship

And we have the following results:P0 = Prob that no one in the queue

Ls= number people waiting in the system

Lq=number people waiting in the queue

Ws= total waiting time in the system

Wq= total waiting time in the queue

How these L, W values are represented

(to p18)

(to p6)

6

Wq, Lq

Ws, Ls

Busy time =

U

Idle time = P0 = 1 -

Their relationships (to p7)

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The Single-Server Waiting Line SystemBasic Single-Server Queuing Formulas

Probability that no customers are in the queuing system:

Probability that n customers are in the system:

Average number of customers in system: and waiting line:

Average time customer spends waiting and being served:

Average time customer spends waiting in the queue:

Probability that server is busy (utilization factor):

Probability that server is idle:

11oP

11 n

n

o

n

n PP

LLq

2

LW11

LWq

1

U

11 UI

L

Note: the process to derive these formulasare based on the “birth-and-death” process

Example(to p8)

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The Single-Server Waiting Line SystemOperating Characteristics for Fast Shop Market Example

Given: = 24 customers per hour arrive at checkout counter,

= 30 customers per hour can be checked out

= (1 - 24/30) = .20 probability of no customers in the system.

= 24/(30 - 24) = 4 customers on ther average in the system

= (24)2/[30(30 -24)] = 3.2 customers on the average in the waiting line

= 1/[30 -24] = 0.167 hour (10 minutes) average time in the system per customer

= 24/[30(30 -24)] = 0.133 hour (8 minutes) average time in the waiting line

= 24/30 = .80 probability server busy, .20 probability server will be idle

1oP

L

2

qL

L

W

1

qW

U

Example:

Then,

(to p9)

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The Single-Server Waiting Line System

Steady-State Operating Characteristics

Because of steady -state nature of operating characteristics:

- Utilization factor, U, must be less than one: U<1,or / <1 and < .

- The ratio of the arrival rate to the service rate must be less than one

or, the service rate must be greater than the arrival rate.

- The server must be able to serve customers faster than the arrival rate in the long run, or waiting line will grow to infinite size.

What if Utilization rate >= 1? (what would happened?)

Changing of the values of , .

Important note:

(to p10)

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• Consider another case where,

– Manager wishes to test several alternatives for reducing customer waiting time:

• 1. Addition of another employee to pack up purchases

• 2. Addition of another checkout counter.

– Which one of the three models that we should deploy?

• Answer:

– Comparing its operating characteristics

• (all three examples with calculation) (to p13)

(to p11)

(to p12)

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The Single-Server Waiting Line SystemEffect of Operating Characteristics on Managerial Decisions

(1 of 3)

- Alternative 1: Addition of an employee (raises service rate from = 30 to = 40 customers per hour)

Cost $150 per week, avoids loss of $75 per week for each minute of reduced customer waiting time.

System operating characteristics with new parameters:

Po = .40 probability of no customers in the system

L = 1.5 customers on the average in the queuing system

Lq = 0.90 customer on the average in the waiting line

W = 0.063 hour (3.75 minutes) average time in the system per customer

Wq = 0.038 hour ( 2.25 minutes) average time in the waiting line per customer

U = .60 probability that server is busy and customer must wait, .40 probability server available

Average customer waiting time reduced from 8 to 2.25 minutes worth $431.25 per week.

Net savings = $431.25 - 150 = $281.25 per week.

= 1 - (24/40)

i.e. (5.75*$75)

(to p10)

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(2 of 3)- Alternative 2: Addition of a new checkout counter ($6,000 plus $200 per week for additional cashier)

=24/2 = 12 customers per hour per checkout counter.

= 30 customers per hour at each counter

System operating chacteristics with new parameters:

Po = .60 probability of no customers in the system

L = 0.67 customer in the queuing system

Lq = 0.27 customer in the waiting line

W = 0.055 hour (3.33 minutes) per customer in the system

Wq = 0.022 hour (1.33 minutes) per customer in the waiting line

U = .40 probability that a customer must wait

I = .60 probability that server is idle and customer can be served.

Savings from reduced waiting time worth $500 per week - $200 = $300 net savings per week.

After $6,000 recovered, alternative 2 would provide $300 -281.25 = $18.75 more savings per week.

(to p10)

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(3 of 3)

Table 13.1Operating Characteristics for Each Alternative System

Figure 13.2Cost trade-offs for service

levels

Decision: very much dependedon manager’s experience because there is difficult to obtain “the”best solution ..

Note: Min cost is not obtained here

(to p14)

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One last note

• The Single waiting line system we studied here can be denoted as: (to p15)

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Models

In this subject, we only consider the following model(s) only:

(M/M/1):(GD/a/a)

Possion arrival rate

Exponential service rate

One service channel General queue disciplinesuch as FCFS

Infinite length

Infinite calling pop

It has a general format like (to p16)

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Type of models

(A/B/C): (D/E/F)

Number of channels(parallel servers)

Arrival distribution

Service distribution Queue capacity,such max ppl in the queue length

Queue discipline, such as FCFS

Size of population

Tutorial Questions (to p17)

Example: M/M/S with finite population

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Tutorial Questions

• Ex: 7, 8, 9 and 12

(END)

End

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19

The Single-Server Waiting Line SystemOperating Characteristics for Fast Shop Market Example

Given: = 24 = 24 = 24 /2=12

= 30 = 40 = 30

= (1 - 24/30) = .20 1-24/40 =0.4 1-12/30=0.6

= 24/(30 - 24) = 4 1.5 0.67

= (24)2/[30(30 -24)] = 3.2 0.9 0.27

= 1/[30 -24] = 0.167 hour (10 minutes) 0.063(3.75 mins) 0.055(3.33min)

= 24/[30(30 -24)] = 0.133 hour (8 minutes) 0.038 (2.25 min) 0.022 (1.33 min)

= 24/30 = .80 probability server busy, .20 0.6 0.4

1oP

L

2

qL

L

W

1

qW

U

Example:1

Then,

(to p9)

Alternative 1 Alternative 2

1 Queuing Analysis Overview What is queuing analysis? - to study how people behave in waiting in line so that we could provide a solution with minimizing.

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