Queuing theory is the mathematical study of waiting lines in systems like transportation, banks, and stores. It was developed in 1903 and is used to predict system performance and determine costs. Queuing models make assumptions like customers arriving randomly and service times being exponentially distributed. They can be applied to situations involving customers like restaurants or manufacturing. The models provide metrics like expected wait times that are used to optimize staffing and inventory levels.

1. By- Dr. B. J. Mohite
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Queuing Theory

2. Queuing Theory
• In 1903 the Danish telecommunication engineer
Agner Krarup Erlang started applying principles of
queuing theory in the area of telecommunications
in predicting and evaluating system performance
• Queuing theory is the mathematical study of
waiting lines, or queues.
• Queuing theory is generally considered a branch of
operations research because the results are often
used when making business decisions to determine
the balance between cost of offering the service
and cost incurred due to delay in offering service.

3. Applications of Queuing Model
Queuing Model can be applied to various situations :
• Commercial Queuing Systems where customers are involved such
as restaurants, banks, super market, airports, ATM, Petrol Pump
etc.
• Very useful in Manufacturing units
• Applicable for the problem of machine breakdown & repairs
• Applicable for the scheduling of jobs in production control
• Applicable for the minimization of traffic congestion at tollbooth
• Provide solution of inventory control problems
• It is a practical operations management technique that is commonly
used to determine staffing, scheduling and calculating inventory
levels.
• To improve customer satisfaction.

4. Assumption in Queuing Model
• Independent arrivals: The customer arrive for service at a
single service facility at random according to Poisson
distribution with mean arrival rate λ.
• Exponential distributions: The service time has exponential
distribution with mean service rate µ.
• The service discipline followed is First Come First Served.
• Customer Behavior is Normal i.e Customers do not leave or
change queues.
• Service facility behavior is Normal
• The calling source has infinite size
• The mean arrival rate is less than the mean service rate
• The waiting space available for customer in the queue is
infinite

5. Limitations of Queuing Model
• The waiting space for the customer is usually limited
• The arrival rate may be state dependent
• The arrival process may not be stationary
• The population of customers may not be infinite and
the queuing discipline may not be FCFS
• Services may not be rendered continuously
• The Queuing system may not have reached the steady
state. It may be, instead, in transient state
• Theoretical solution may either prove intractable or
insufficiently informative to be useful.

6. What is Waiting Time Cost &
Idle Time Cost?
• The cost of waiting customers include either
the indirect cost of lost business or direct
cost of idle equipment and persons.
• The cost of idle service facilities is the
payment to be made to the servers for the
period for which they remain idle.

7. What is Transient & Steady State
of the System?
• If the operating characteristics vary with time then
it is said to be transient state of the system. OR
When the operating characteristics are dependent
on time, it is said to be a transient state.
• If the behavior becomes independent of its initial
conditions and of the elapsed time is called Steady
State condition of the system. OR When the
operating characteristics are independent of time,
it is said to be a steady state.

8. Customer’s Behavior
• Balking: If a customer decides not to enter the
queue since it is too long, has no time to wait, no
space to stand etc. is called Balking.
• Reneging: If a customer enters the queue but
after sometimes loses patience and leaves the
queue then it is called Reneging.
• Jockeying: When there are 2 or more parallel
queues and the customers move from one queue
to another is called Jockeying.

9. Basic Structure of Queuing Model

10. Elements of a Basic Queuing System

11. 1. The Calling Population
– The population from which customers/jobs
originate
– The size can be finite or infinite (the latter is
most common)
– Can be homogeneous (only one type of
customers/ jobs) or heterogeneous (several
different kinds of customers/jobs)

12. 2. Arrival Process
• In what pattern do jobs / customers arrive to the
queuing system?
– Determines how, when and where customer/jobs
arrive to the system
– Random or Batch arrivals?
– Finite population?
– Finite queue length?
• Poisson arrival process often assumed
– Many real-world arrival processes can be modeled
using a Poisson process

13. 3. Service Process
• How long does it take to service a job or customer?
– Distribution of arrival times?
– Service center (machine) breakdown?
• Exponential service times often assumed
– Works well for maintenance or unscheduled service
situations

14. 4. Service Mechanism
– Can involve one or several service facilities with one or
several parallel service channels or servers like
 Single Queue Single-server
 Single Queue Multiple-server
 Multiple Queue Single-server
 Multiple Queue Multiple-server
– The service provided by a server is
characterized by its service time
– Most analytical queuing
models are based on the assumption
of exponentially distributed service times.

15. 5. Queue Discipline
• Specifies the order by which jobs in the queue are
being served.
– Most commonly used principle is First Come
First Served (FCFS)
– Other rules are,
Shortest Processing Time (SPT)
Earliest Due Date (EDD)
Priority (jobs are in different priority
classes)
• FCFS default assumption for most models

16. Kendall Notations
• Commonly used notation principle: (a/b/c):(d/e/f)
– a = The inter arrival time distribution (Poisson)
– b = The service time distribution (Exponential)
– c = The number of parallel servers
– d= Queue discipline
– e = maximum number (finite/infinite) allowed in the system
– f = size of the calling source(finite/infinite)
• Commonly used distributions
– M = Markov which implies that number of arrivals or departures
in time t.
– D = Deterministic distribution
– G = General distribution
• Example: M/M/c
– Queuing system with exponentially distributed service and inter-
arrival times and c servers

17. 1. No. of Customers in the system = (n)
2. No. of service channels = (s)
3. Max no. of customers allowed in system = (N)
4. Average customer arrival rate or average no. of
arrivals per unit of time in queuing system. = (λ)
5. Average service rate or average no. of customers
served per unit time at the place of service = (µ)
6. Traffic intensity or Service utilization factor
(Probability that the service facility is busy/there is
at least one customer in system) ρ= λ/ µ
7. Average (expected) number of customers in the
system waiting and being served. Ls= λ/ (µ- λ)
8. Average (expected) number of customers waiting
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Queuing Models Notations & Formulae’s

18. Queuing Models Notations & Formulae’s
9. Average (expected) time a customer spends in the
system waiting and being served. Ws= 1/ (µ- λ)
10.Average (expected) time a customer spends waiting in
the waiting line or queue. Wq= λ/µ(µ- λ) or =Ws-(1/µ)
11.Probability no customers in the system/ system is idle
P0=1- ρ or = 1-(λ/µ)
9. Probability n customers in the system being served &
waiting in queue Pn=P0(λ/µ)n
10.Probability that the queue length ≥ k / no. of customers
exceeds k P(n>k)= (λ/µ)k+1
11.Average (expected) waiting time of customer in the
queue who waits for service. Ww= 1/ (µ- λ)
12.Average (expected) no. of customer in a Non-empty
queue. i.e. queue length. L = µ/(µ- λ)

19. Single Server Queuing Model (M/M/1:∞/FIFO)
In this situation the customers arriving in a single queue are
served by a single server. Assumptions of this Model
Infinite Calling Populations or Number of arrivals per unit
time
The arrival process is Poisson with an expected arrival rate λ
The queue configuration is a single queue with possibly
infinite length i.e. No reneging or balking
The queue discipline is FIFO
The service mechanism consists of a single server with
exponentially distributed service times
The waiting space for the customers in queue is infinite
Mean arrival rate (λ) < Mean service Rate (µ)