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basics of stochastic and queueing theory
1. 1. APOORVA GUPTA(11972)
2. JYOTI ( 11973)
M.Sc. (Mathematical Sciences- STATISTICS)
2014-2015
Under the supervision of
Dr. Gulab Bura
Department of Mathematics and Statistics,
AIM & ACT
Basic Terminologies related
to Queuing theory
2. Content
• Stochastic process
• Markov process
• Markov chain
• Poisson process
• Birth death process
• Introduction to queueing theory
• History
• Elements of Queuing System
• A Commonly Seen Queuing Model
• Application
• Future plan
• References
3. Stochastic process
Definition : A stochastic process is family of time
indexed random variable where t belongs to
index set . Formal notation , where I is an
index set that is subset of R.
Examples :
• No. of telephone calls are received at a switchboard.
Suppose that is the r.v. which represents the
number of incoming calls in an interval (0,t) of
duration t units.
• Outcomes of X(t) are called states .
}:{ ItXt
tX
tX
4. • Discrete time , Discrete state space
• Discrete time , Continuous state space
• Continuous time , Discrete state space
• Continuous time, Continuous state space
Stochastic process is classified in four categories on
the basis of state space and time space
5. EXAMPLES
• Discrete time, Discrete state space : number of
accidents at 9 pm, 10 pm, 11 pm etc.
• Continuous –time ,Discrete state space : Number
of accidents on a highway at an interval of time
say 9 pm to 10 pm.
• Discrete time ,continuous state space :age of
people in a city at a particular year say 2001,
2002, 2003 etc.
• Continuous time , Continuous state space :age of
people from March 2013 to march 2014.
6. MARKOV PROCESS
If is a stochastic process such that ,given the
values of X(t) , t >s ,do not depend on the values of X(u) ,u<s,
then the process is said to be a Markov process.
• A definition of such a process is given as :
If for ,
the process is a markov process .
}),({ TttX
tttt n ...21
})(,...,)(|)(Pr{ 11 nn xtXxtXxtX
})(|)(Pr{ nn xtXxtX
}),({ TttX
)}(|)(Pr{)}(),(|)(Pr{ sXtXuXsXtX
7. Markov chain :
A discrete parameter Markov process is known as
Markov chain.
Definition : The stochastic process is
called a markov chain ,if , for
= It is the probability of transition from the
state ‘j’ at (n-1) th trial to the state k at nth trial .
,......}2,1,0,{ nXn
Njjkj n 11,...,,,
},...,,|Pr{ 10121 nnnn jXjXjXkX
jknn pjXkX }|Pr{ 1
jkp
8. Example of markov chain
Example : random walk between two barriers .
Homogeneous chain:
the transition probability independent of n, the
markov chain said to homogeneous.
Two types of markov chain :
• Discrete time markov chain
• Continuous time markov chain
}|Pr{ 1
jXkXp nnjk
9. • A stochastic process {N(t), t>0} with discrete
state space and continuous time is called a
poisson process if it satisfies the following
postulates:
1. Independence
2. Homogeneity in time
3. Regularity
The Poisson Process
10. 10
• Independence:
N( t+h)-N(t), the number of occurrence in the
interval (t,t+h) is independent of the number of
occurrences prior to that interval.
• Homogeneity in time:
pn(t) depends only on length ‘t’ of the interval and is
independent of where this interval is situated i.e.
pn(t) gives the probability of the number of
occurrences in the interval (t1 , t+t1 ) (which is of
length ‘t’) for every t1 .
Postulates for Poisson Process
11. Regularity In the interval of infinite small length
h, the probability of exactly one occurrence is
λh+o(h) and that of occurrence of more than one is
of o(h).
p1 (h)= λh+o(h)
)(
2
hOhp
k
k
)(10 hohhP
λ= rate of arrival
Probability of zero arrival
12. • Pure birth process is originated from Poisson process
in which the rate of birth depends on number of
persons.
Pure Birth Processes
P1 (t)=λnh+o(h)
Pn (t) = o(h) , k>=2
P0(t)= 1- λnh+o(h),
13. Birth-and-Death Process
• We consider pure birth process where
Here only births are considered, further deaths are also
considered as possible so,
Collectively for n>=k, equations are called as Birth and Death
process
P1 (t)=λn(h)+o(h) , k=1
Pn (t) = o(h), k>=2
P0 (t)= 1- λn(h)+o(h),k=0
q1 (t)=μn(h)+o(h) , k=1
qn (t) = o(h), k>=2
q0 (t)= 1- μn(h)+o(h),k=0
14. • Note: The foundation of many of the most
commonly used queuing models
Birth – equivalent to the arrival of a customer
Death – equivalent to the departure of a served
customer
15. 15
A Birth-and-Death Process Rate Diagram
0 1 n-1 n
0 1 2 n-1 n n+1
1 2 n n+1
n = State n, i.e., the case of n customers in the system
Excellent tool for describing the mechanics of a Birth-and-
Death process
16.
17. Introduction of Queueing theory
• Queueing theory is the mathematical study of
waiting lines, or queues.
• In queueing theory a model is constructed so that
queue lengths and waiting times can be predicted.
• Queueing theory is generally considered a branch
of operations research because the results are often
used when making business decisions about the
resources needed to provide a service.
18. DEFINITION
• A queue is said to occur when the rate at which the
demand arises exceeds the rate at which service is
being provided.
• It is the quantitative technique which consists of
constructing mathematical models for various types
of queuing systems.
• Mathematical models are constructed so that queue
lengths and waiting times can be predicted which
helps in balancing the cost of service and the cost
associated with customers waiting for service
19. HISTORY
• Queueing theory’s history goes back
nearly 105 years.
• It is originated by A. K. Erlang in 1909, in
Danish when he is at his work place.
• Agner Krarup Erlang, a Danish engineer
who worked for the Copenhagen
Telephone Exchange, published the first
paper on what would now be called
queueing theory in 1909.
• He modeled the number of telephone
calls arriving at an exchange by a Poisson
process.
20. • Johannsen’s “Waiting Times and Number of Calls”
seems to be the first paper on the subject.
• But the method used in this paper was not
mathematically exact and therefore, from the point
of view of exact treatment, the paper that has
historic importance is A. K. Erlang’s, “The Theory of
Probabilities and Telephone Conversations”
• In this paper he lays the foundation for the place of
Poisson (and hence, exponential) distribution in
queueing theory.
21. • His most important work, Solutions of Some
Problems in the Theory of Probabilities of
Significance in Automatic Telephone Exchanges , was
published in 1917, which contained formulas for loss
and waiting probabilities which are now known as
Erlang’s loss formula (or Erlang B-formula) and delay
formula (or Erlang C-formula), respectively.
Agner Krarup Erlang, 1878–1929
22. Elements of Queuing System
Service process
Queue Discipline
Number of
servers
Elements of Queuing
Models
Arrival process
Customer’s
Behavior
System capacity
23. 1. Arrival Process
Arrivals can be measured as the arrival rate or the interarrival
time (time between arrivals).
• These quantities may be deterministic or stochastic (given by a
probability distribution).
• Arrivals may also come in batches of multiple customers,
which is called batch or bulk arrivals.
• The batch size may be either deterministic or stochastic.
Interarrival time =1/ arrival rate
24. 2.Customer’s Behaviour
• Balking: If a customer enters a system and
decides not to enter the queue since it is too
long is called Balking.
• Reneging: If a customer enters the queue but
after sometimes loses patience and leaves it is
called Reneging.
• Jockeying: When there are 2 or more parallel
queues and the customers move from one
queue to another to change his position is called
Jockeying.
25. 25
3. Service Process
• Service Process determines the customer service
times in the system.
• As with arrival patterns, service patterns may be
deterministic or stochastic. There may also be
batched services.
• The service rate may be state-dependent. (This is the
analogue of impatience with arrivals.)
Note that there is an important difference between
arrivals and services. Services do not occur when
the queue is empty.
26. 26
4. Number of Servers
Single Server Queue:
Multiple Server Queue
28. 5. Queue Discipline
• FIFO- First in First out
Or FCFS- First Come First Serve
• LIFO-Last in First out
• SIRO- Service in Random order
• Priority based
33. 6.System capacity
• The capacity of a system can be finite or infinite.
There are many queuing systems whose capacity is
finite. In such cases problem of balking arises. It is
known as forced balking. In forced balking customer
needs the service but due to the problem of limited
capacity of the system the customer leaves the
system.
34. System Customers Server
Reception desk People Receptionist
Hospital Patients Doctors
Airport Airplanes Runway
Road network Cars Traffic light
Grocery Shoppers Checkout station
Queuing examples
35. A Commonly Seen Queuing Model
• Service times as well as inter arrival times are assumed
independent and identically distributed
– If not otherwise specified
• Commonly used notation principle: A/B/C
– A = The inter arrival time distribution
– B = The service time distribution
– C = The number of parallel servers
• Commonly used distributions
– M = Markovian (exponential) - Memory less
– D = Deterministic distribution
– G = General distribution
• Example: M/M/1
– Queuing system with exponentially distributed service and
inter-arrival times and 1 server
36.
37. Examples of Different Queuing Systems
• Arrival Distribution: Poisson rate (M) tells you to
use exponential probability
• Service Distribution: again the M signifies an
exponential probability
• 1 represents the number of servers
M/M/1: The system consists of only one server. This queuing
system can be applied to a wide variety of problems as any
system with a very large number customers.
M/M/1
38. M/M/1 queueing systems assume a Poisson arrival
process.
This assumption is a very good approximation for
arrival process in real systems that meet the
following rules:
•The number of customers in the system is very
large.
•Impact of a single customer on the performance of
the system is very small, i.e. a single customer
consumes a very small percentage of the system
resources.
•All customers are independent, i.e. their decision to
use the system are independent of other users.
39. Application of Queuing Theory
• Telecommunication.
• Traffic control.
• Determine the sequence of computer
operations.
• Predicting computer performance.
• Health service ( e.g. Control of hospital bed
assignments).
• Airport traffic, airline ticket sales.
• Layout of manufacturing systems.
