Waiting Line
Car Wash, ATM, Machine Repair - people form line
Lines could be of people and items
Line related to human being have emotions related. So, there is a direct concern. Components
of a queue
1.
2.
3.
4.
Arrival
Queue – actual waiting line
Get Service – service facility
Departure
For better service, you need more cost. The graph will be diagonal straight line. Cost of waiting
time graph will go downwards. Total expected cost will be U shaped. Middle point will be the
optimum value.
In management science terminology, a waiting line is also known as a queue, and the body of
knowledge dealing with waiting lines is known as queueing theory.
In the early 1900s, A. K. Erlang, a Danish telephone engineer, began a study of the congestion
and waiting times occurring in the completion of telephone calls.
Waiting line models consist of mathematical formulas and relationships that can be used to
determine the operating characteristics (performance measures) for a waiting line. Operating
characteristics of interest include these:
1. The probability that no units are in the system (i.e., the system is idle)
2. The average number of units in the waiting line
3. The average number of units in the system (the number of units in the waiting line
plus, the number of units being served)
4. The average time a unit spends in the waiting line
5. The average time a unit spends in the system (the waiting time plus the service time)
6. The probability that an arriving unit has to wait for service
15.1 Structure of a Waiting Line System
Single-Server Waiting Line
One person per time served by one person. As there is only one person serving, there will be
one line.
Distribution of Arrivals
Waiting line situations, the arrivals occur randomly and independently of other arrivals, and we
cannot predict when an arrival will occur. In such cases, analysts have found that the Poisson
probability distribution provides a good description of the arrival pattern.
Service Time
The service time is the time a customer spends at the service facility once the service has
started.
If the probability distribution for the service time can be assumed to follow an exponential
probability distribution, formulas are available for providing useful information about the
operation of the waiting line.
60 minute e 60 ta = mean would be 1 customer/minute
P(service time < 0.5 min.) = 1 - e-1(0.5) = 1 - 0.6065 = 0.3935
P(service time < 1.0 min.) = 1 - e-1(1.0) = 1 - 0.3679 = 0.6321
P(service time < 2.0 min.) = 1 - e-1(2.0) = 1 - 0.1353 = 0.8647
Queue Discipline
For the Burger Dome waiting line, and in general for most customer-oriented waiting lines, the
units waiting for service are arranged on a first-come, first-served basis; this approach is
referred to as an FCFS queue discipline.
Whole chapter follows this.
Steady State Operation
When the Burger Dome restaurant opens in the morning, no customers are in the restaurant,
and the characteristics of the waiting line system fluctuate depending on realized arrival and
service times. Gradually, activity builds up to a normal or steady state. The beginning or startup
period is referred to as the transient period. The transient period ends when the system
reaches the normal or steady-state operation.
Waiting line model Steady State Operation
Generally, service rate improvements are obtained by making either or both of the following
changes:
1. Increase the service rate by making a creative design change or by using new technology.
2. Add one or more servers so that more customers can be served simultaneously.
Multiple Server Waiting Line
A multiple server waiting line consists of two or more servers that are assumed to be identical
in terms of service capability. For multiple-server systems, there are two typical queueing
possibilities: (1) arriving customers wait in a single waiting line (called a “pooled” or “shared”
queue) and then move to the first available server for processing, or (2) each server has a
“dedicated” queue and an arriving customer selects one of these lines to join (and typically is
not allowed to switch lines).
Formulas that can be used to determine the steady-state operating characteristics for a
multiple-server waiting line. These formulas are applicable if the following conditions exist:
1. The arrivals follow a Poisson probability distribution.
2. The service time for each server follows an exponential probability distribution.
3. The service rate λ is the same for each server.
4. The arrivals wait in a single waiting line and then move to the first open server for service.
Little’s Flow Equations
John D. C. Little showed that several relationships exist among these four characteristics
and that these relationships apply to a variety of different waiting line systems. Two of
the relationships, referred to as Little’s flow equations, are
Where,
Lq = the average number of units in the waiting line
L = the average number of units in the system
Wq = the average time a unit spends in the waiting line
W = the average time a unit spends in the system
The average time in the system =
For a system with a service rate μ, the mean service time is 1/μ.
THE GENERAL SHAPE OF WAITING COST, SERVICE COST, AND TOTAL COST CURVES IN WAITING
LINE MODELS.
Kendall Three Symbol Notation
D. G. Kendall suggested a notation that is helpful in classifying the wide variety of different
waiting line models that have been developed. The three-symbol Kendall notation is as follows:
A/B/k = M/G/1
were
A denotes the probability distribution for the arrivals
B denotes the probability distribution for the service time
k denotes the number of servers
Class Discussion
Service cost increase, waiting cost decrease.
Waiting time cannot be zero because that is never the objective. Minimizing the total expected
cost is the objective. Waiting cost + cost of providing service is total expected cost.
Estimated costs: Lost time, Lost of goodwill, Lost sales so they are difficult to measure
Accountants estimate these costs
Average figures are considered.
Demurrage ships
Stevedores help to shift goods
Poisson Distribution no relationship with arrivals
Passport Office, Vaccination Multiphase
Arrival is Poisson because it is normal phenomena
M/M/1 is the most common one, M/M/2 is also used
Assembly Line Balancing
Station is made by one person.
Balancing
Dish washing vs Dish Drying – 4 Dish cleaner vs 1 Dish Dryer – Imbalanced
One person alone can complete their task A,B,C. They do not require previous output.
Total time 510 s.
One person alone would require 510 per calculator. Time stamp 0,510,1020
Time difference between two output – cycle time
High time difference = Low productivity per day
8 hours per day shift means, a person would produce 56.47 ~ 56
If one person is assigned for one specific task, it would require 11 individuals.
Here, cycle time would be 75s. The maximum time.
Explanation:
A  10s
B  40s
C  10s
Unit
1
2
3
A
Start
0
10
20
Difference = 40s
End
10
20
30
B
Start
10
50
90
End
50
90
130
C
Start
50
90
130
End
60
100
140
If the cycle time is 75s, they can produce 384.
This one assembly line can produce this much. If someone requires 500 calculator per day,
make another line.
Labor cost per production:
Per hour per person $100
For one person, labor cost per production is = 800/56 = 14.28
For 11 people, labor cost per production is = 8800/384 = 22.9
Difference is because of idle time.
Desired output
For 200 units, cycle time would be = 8*60*60/200 = 144s
We would need = 510/144 = 4 people/station
Diagram
AON – Activity on Node
At first, assign the free jobs: A,B,C
Relation will be straight arrow 
Movement will be left to right 
Try to avoid line cross
Table Format
Longest Time Rule: To reduce idle time
This would require 5 stations. Tf
Station 1
C
B
Station 2
A
Free Job method
A–6
B–7
C–1
D–4
E–4
F–4
G–3
Station 1
B
A
D
Station 2
F
E
H
Station 3
G
I
C
Station 4
J
K
Monte Carlo Simulation
Monte Carlo is a city of gambling.
Results from the simulation will be close to the actual value.
For writing multiple values at once
Fill > Step value 1 > Stop Value 100
Game Theory
For Zero Sum Payoff Matrix, the everything is considered for the row player. One person’s gain
= One person’s loss = 10+ (-10) = 0
For 3, X is getting $3 and Y is losing $3
For -6, X is losing $6 and Y is gaining $6
Pure Strategy: Here both of the parties will shift due to strategical approaches
Saddle Point: There is a specific solution to the problem which is the saddle point
As we can see, X is likely to go for 3 where both of the uses radio. However, Y would then shift
to -6 as there, it will gain more and X will face a loss. From -6, X will go for 1 as that wont give it
$1. Y wont be able to shift from here because if it shifts to the only available option 2, it will
face a loss of $2, which is more than the present one. So, 1 will be the saddle point.
Therefore, this is a pure strategy
Notice that X follows maximin as Y would like to select -6 due to rational decision making. So,
the minimum possible values for X1, X2 are -6, 1 for X. It will choose 1 which is the maximum of
these values.
Y follows Minimax. For Y1, the maximum value is 3. For Y2, the maximum value is 1. So,
minimum of these values would be 1.
If there is no saddle point, there would be mixed strategy
/
Dominant Strategy: The one that dominates
Dominated Strategy: The one that got dominated
Here, if B1 is fixed, between A1 and A3, A1 is better. Same goes if B2, B3 are fixed. So, A1 would
be first choice of player A. Therefore, it is the dominant strategy and A3 would be the
dominated one. Dominated Strategies are disregarded
Here, 3 is Player A’s value. 9 is player B’s value.
If we start from the R cell, UR cell, it will not be a likely play as A can make more profit by going
below. It will move to DR but this will be a likely play. It will not be able to move to DL. If both
parties want, they can stay here. Moving to DL would decrease their profit. Or they can move to
UL. That is also a likely play.
It will keep moving. This is Prisoner’s Dilemma.
Clyde confess korle Bonnier jonno Confess kora bhalo. Bonnie confess korle Clyde er jonno
confess kora bhalo.
Inefficient solution of this table is (-5,-5)
Here, there is a Pareto Optimality. This means if one of them gets better facility, another one
would suffer and cant remain the same. Both of them wont get better facility together.
Nash Equilibrium is not usually Pareto Optimal.
Sequential Market
New Player may enter or stay out
Old Player may fight for price or accommodate entry
For Daily Star, Middle East is the dominant strategy because whatever option independent
chooses, middle east brings more profit.
For Independent, Middle East is the dominant strategy because whatever option daily star
chooses, middle east brings more profit.
For Pepsi, High becomes a dominant strategy because High is better in 2 aspects and 2nd better
in one aspect.
Saddle point is 225\195