This document provides an introduction to queuing models and simulation. It discusses key characteristics of queuing systems such as arrival processes, service times, queue discipline, and performance measures. Common queuing notations are also introduced, including the widely used Kendall notation. Examples of queuing systems from various applications are provided to illustrate real-world scenarios that can be modeled using queuing theory.

1. Queuing Models
Unit 4

2. Contents
• Characteristics of queuing systems
• Queuing notation
• Simulation Examples:
• Queuing
• Inventory System
2

3. Introduction
• Simulation is often used in the analysis of queuing models
• Queueing models whether solved mathematically or analysed
through simulation, provide the analyst a powerful tool for
designing and evaluating the performance of queuing systems.
• Measures of system performance include
• Server utilization (percentage of time a server is busy)
• Length of waiting lines
• Delays of customers
• Simple queueing model is shown below
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4. Simple queueing model 4

5. Simple queueing model
• Customers arrive from time to time and join a queue (waiting line)
• They are eventually served and then finally they leave the system
• Customers refers to any type of entity that can be viewed as
requesting a service from a system
• Eg. : service facilities, production systems, repair and
maintenance facilities, communications and computer systems and
transport and material handling systems
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6. Characteristics of queuing system
• The key elements of queuing system are the customers and server
• Customer refers to anything that arrives at a facility and requires
service
• Eg. people, machines, truck, mechanics, patients, pallets, airplanes ,
email, cases , orders , or dirty clothes
• Server refers to anything that provides the requested service.
• Eg. receptionists, repair personnel, mechanics, medical personnel,
automatic storage and retrival machines such as cranes, runways at airport,
automatic packers, order pickers, washing machines, CPU in computers
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7. Examples of
queueing
system
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8. Characteristics of queuing system
• Calling population
• System capacity
• The Arrival Process
• Queue Behavior and Queue Discipline
• Service Times and the service Mechanism
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9. The calling population
• The population of potential customers, is referred as the calling
population which may be finite or infinite
• Consider the following scenario to understand the terms calling
population, customers and server
• Consider the personal computers of the employees of a small
company that are supported by the IT staff of three technicians
• When a computer fails, needs new software etc, it is attended by
one of IT staff.
• Computers are the customers, IT staff is a server and calling
population is finite here, consists of the personal computers at the
company.
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10. The calling population
• The systems with large population of potential customers, the
calling population is assumed to be infinite.
• The difference between the finite and infinite calling population is
how the arrival rate is defined.
• In infinite calling population, the arrival rate is not affected by
the number of customers left the calling population and joined the
queuing system
• In finite calling population, the arrival rate to the queueing
system does depend on the number of customers being served and
waiting.
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11. The calling population
• The arrival rate defined as the expected number of arrivals in the
next unit of time
• Eg. Consider a hospital with 5 patients assigned to a single nurse.
• When all the patients are resting , the nurse is idle hence the
arrival rate is maximum since any of the patients can call nurse
for assistance next instant
• When all the 5 patients have called the nurse then arrival rate is
zero i.e. no arrival is possible until the nurse finishes with a
patient.
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12. System Capacity
• The limit to the number of customers that may be in the waiting
line or system
• Eg. Automatic car wash might have room for 10 cars to wait in a
line to enter into the mechanism
• When the system capacity is reached, the new customers
immediately joins the calling population
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13. System capacity
• When the system is with limited capacity, distinction is made
between arrival rate and effective arrival rate
• Arrival rate  number of arrivals per time unit
• Effective arrival rate  the number who arrive and enter the
system per unit time
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14. The arrival process
• The arrival process for infinite population models is usually
characterized in terms of interarrival times of successive
customers.
• May occur in scheduled times or random times
• Customers may arrive one at a time or in batches.
• The bacths may be of constant size or of random size.
• Most important model for random arrivals is Poisson arrival
process.
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15. • If An represents the interarrival time in between customer n-1 and
customer n then for Poisson arrival process An is exponentially
distributed with the mean 1/λ time units.
• The arrival rate is λ customers per unit time.
• Eg. Arrival of people for resturants, banks, arrival of telephone
calls at call center, the arrival of demands, orders for a service or
product arrival of failed components machines for a repair
facility.,
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16. • Second type of arrivals is scheduled arrivals, such as patients to a
doctor’s office or scheduled airline flight arrivals to an airport
• Third type of arrival is when at least a customer is assumed to
always to be present in the queue so that the server is never idle
because of lack of customers.
• In case of finite population models arrival process is classified as
pending and not pending
• Customer is defined as pending when customer is outside the
queuing system and a member of calling population
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17. • Customer is defined as not pending when the customer gets served
by the server
• Eg. In a hospital the patients are pending when they are resting
and becomes not pending the instant they call for the nurse
• Runtime is defined for every customer i.e. length of time from
departure from the queuing system until the next customer arrives
into the queue.
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18. Arrival process for a finite population model 18

19. Queuing behaviour and queuing discipline
• Queue behaviour refers to the actions of the customers while in a
queue waiting for a service to begin
• Incoming customers will
• Balk – leave when they see that the line is too long
• Renege- leave after being in the line when they see
the line is moving to slow
• Jockey- move from one line another if they think
they have chosen a slow line
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20. • Queue discipline refers to logical ordering of customers in a queue
and determines which customers will be chosen for service when a
server becomes free
• Some queue disciplines include FIFO, LIFO, service in random
order (SIRO), shortest processing time first(SPT), service according
to priority (PR)
• In FIFO, the service begin in the same order as arrivals but the
customers can leave the system because of different length
service times
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21. Service times and service mechanism
• Service times of successive arrivals are denoted by S1,S2,S3 …
• They may be constant or random.
• Customers can have same service times for a class or type of
customers
• Some times, different customers can also have different service
time distributions
• Service time may depend on time of day or upon the length of
waiting line
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22. • Queueing system consists of number of service centers and
interconnecting queues.
• Parallel service mechanisms are either single server, multiple
server or unlimited server
• Self service facility is usually characterized as having unlimited
number of servers.
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23. Eg. Warehouse
• Customers may either serve themselves or wait for one of three
clerks and then finally leave after paying at a single cashier.
• The system flow is shown in the following figure
• The subsystem, consisting of queue 2 and service center 2 is
shown in the figure
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24. Discount warehouse with three service
centers 24

25. Service center 2, with c = 3 parallel servers 25

26. Ex. Candy manufacturer
• Has a production line that consists of three machines separated by
inventory in process buffers
• First machine makes and wraps the individual pieces of candy
• Second packs 50 pieces in a box
• Third machine seals and wraps the box.
• The inventory buffers have a capacity of 1000 boxes each
• Machine 1 shuts down whenever its inventory buffer fills to
capacity and machine 2 shuts down whenever its buffer empties.
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27. Candy production line 27

28. Queuing notation
• Recognizing the diversity of queuing systems, Kendall proposed a
national system for parallel server systems which has been widely
adopted.
• The model is based on the format A/B/c/N/K. these letters
represent the following system characteristics:
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A the inter arrival time distribution
B the service time distributions
C the number of parallel servers
N the system capacity
K the size of the calling population

29. Common symbols for A & B include
• M (exponential or Markov)
• D ( Constant or deterministic)
• Ek (Erlang of order k)
• PH (phase-type)
• H ( hyper exponential)
• G ( arbitrary or general)
• G1 ( general independent)
• Eg: M/M/1/∞/∞ indicates a single- server system that has
unlimited queue capacity and an infinite population
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30. 30

31. Long-Run measures of performance of
queuing systems
• Time Average Number in system ( L )
• The number of customers in a queue (LQ)
• Average Time Spent in System per Customer ( w )
• The conservation Equation: L = λw
• Server Utilization
• Costs in queuing problems
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32. Time average number in system (L) 32

33. Time Average Number in System L
• Consider a queuing system over a period of Time T, & let L(t)
denote the no. of customers in the system at time t. let Ti denote
the total time during [0,T] in which the system contained exactly
i customers. The time –weighted-average number in a system is
defined by
•
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34. • Many queuing systems exhibit a certain kind of long-run stability in
terms of their average performance . for such condition, the long
run time-average number in system , with probability 1 can be
given as
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35. The number of customers in a queue (LQ) 35

36. 36

37. Average Time Spent in System Per
Customer w
• If the simulation is done for a period of time , say T, then record
the time each customer spend in the system during [0,T], say
W1,W2…… WN where N is the number of arrivals during [0,T].
The average time spent in system per customer, called average
system time given as, where w is called long run average system
time.
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38. • If the system under consideration is the queue alone, then the
equation can be written as
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39. example
• For the system history shown in figure 1 N=5 customers. The
system has a single server and FIFO queue discipline.
• Arrivals occurs at the rate of 0,3,5,7 and 16.
• Departure time 2,8,10,14,20
• Find the average time spent in system per customer
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40. figure 40

41. The conservation Equation : L = λW
• Consider a system with N= 5 arrivals in T=20 time units and thus
the observed arrival rate was λ = N/T.
• The relationship between L,λ, W is not coincidental.
• It holds for almost all queuing systems or subsystems regardless of
the number of servers, the queue discipline, or any other special
circumstances allowing T∞ and N ∞ equation becomes
L = λw, where λ is the long-run average arrival rate and the
equation is called conservation equation.
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42. • It says that “ the average number of customers in the system at an
arbitrary point in time is equal to the product of average number
of arrivals per time unit, times the average time spent in the
system.
• The total system time of all customers is given by the total area
under the number-in-system function L(t)
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43. Server Utilization
• Defined as the proportion of time that a server is busy. Long –run
server utilization is denoted by ρ. For the systems that exhibit
long run stability
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44. M/M/1 queue
• M/M/1 queue will often be a useful approximate model when
service times have standard deviations approximately equal to
their means.
• The different steady state parameters can be calculated by
substituting σ² = 1 / μ² in the steady state parameter values of
M/G/1 queueing model
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45. 52

46. problem
• The inter arrival times and service times at a single –chair saloon
have been shown to exponentially distributed. The values of λ and
μ are 2 per hour and 3 per hour respectively. For this M/M/1 queue
determine
1. the time average number of customers in the system
2. The average time an arrival spends in system
3. The average time the customer spends in the queue
4. The time average number in the queue.
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47. Question bank
• Explain the characteristics of queuing system
• Explain the queueing notation A/B/C/N/K with an example
• Explain the steady state parameters of M/M/1 queue
• Explain Long-Run measures of performance of queuing systems
• Give all the queueing notation of parallel server system
• Write short note on network of queues
• Dump truck and inventory system problems
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48. Unit 5 questions
• Explain a single server queue simulation in java
• Explain the event schedule algorithm and list processing operation
• Write the GPSS block diagram for single server queue simulation
• What is boot strapping ? Explain time advance algorithm
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49. End of unit 4
Thank you 