Queuing theory

Seminar
on
INTRODUCTION TO QUEUING THEORY
Submitted to: Submitted by:
Dr. Harikesh Singh Shiwangi Yadav
Er. No.: 142107

• Queuing
• Queuing Notation
• Rules For All Queues
• Little’s Law
• Types Of Stochastic Processes

 A queue is a waiting line of customers requiring service from
one server to another and it is a mathematical study of waiting
lines or queues.
 It is extremely useful in predicting and evaluating the system
performance.
 There are some applications of queuing like Traffic control ,
Health service , Ticket sales etc.
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1. Arrival Process: If the students arrive at times t1,t2,...,tj
the random variables τj = tj – tj-1 are called the interarrival
times. It is generally assumed that τj form a sequence of
Independent and Identically Distributed random variables.
2. Service Time Distribution: We also need to know the
time each student spends at the terminal. This is called the
service time. It is common to assume that the service times
are random variables, which are IID.
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3. Number of Servers: The terminal room may have one or
more terminals, all are considered part of the same queuing
system since they are all identical, and any terminal may be
assigned to any student.
If all the servers are not identical, they are usually divided
into groups of identical servers with separate queues for each
group.
4. System Capacity: The maximum number of students who
can stay may be limited due to space availability and also to
avoid long waiting times. This number is called the system
capacity.
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5. Population Size: The total number of potential students
who can ever come to the computer centre is the
population size.
6. Service Discipline: The order in which the students are
served is called the service discipline. The most common
discipline is First Come First Served (FCFS). Other
possibilities are Last Come First Served (LCFS) and Last
Come First Served with Preempt and Resume (LCFS-
PR).
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There are some key variables used in queueing analysis are
as follow:
 τ = interarrival time, the time between two successive
arrivals.
 λ = mean arrival rate.
 s = service time per job.
 m = number of servers.
 μ= mean service rate per server.
 n= number of jobs in the system. This is also called
queue length.
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 nq = number of jobs waiting to receive service.
 ns = number of jobs receiving service.
 r = response time.
 w = waiting time.
There are a number of relationships among these variables
that apply to G/G/m queues.
1. Stability Condition: If the number of jobs in a system
grows continuously and becomes infinite, the system is
said to be unstable. For stability the mean arrival rate
should be less than the mean service rate is
λ > mμ
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2. Number in System versus Number in Queue: The
number of jobs in the system is always equal to the sum of
the number in the queue and the number receiving service:
n=nq + ns
n, nq, and ns, are random variables.
The mean number of jobs in the system is equal to the
sum of the mean number in of job in the queue and the
mean number receiving service.
E[n]=E[nq] + E[ns]
If the service rate of each server is independent of the
number in the queue, we have
Cov(nq,ns) = 0 and Var[n] = Var[nq] + Var[ns]
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3. Number versus Time: If jobs are not lost due to
insufficient buffers, the mean number of jobs in a system
is related to its mean response time as follows:
Mean number of jobs in system = arrival rate × mean
response time.
Mean number of jobs in queue = arrival rate × mean
waiting time.
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4. Time in System versus Time in Queue: The time
spent by a job in a queueing system.
r = w + s
r, w, and s are random variables
The mean response time is equal to the sum of the mean
waiting time and the mean service time.
E[r] = E[w] + E[s]
If the service rate is independent of the number of jobs in
the queue, we have
Cov(w,s) = 0 and Var[r] = Var[w] + Var[s]
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 Little’s law, which was first proven by Little in 1961.
 The most commonly used theorems in queuing theory is
Little’s law, which allows us to relate the mean number of
jobs in any system with the mean time spent in the system as
follows:
Mean number in the system = arrival rate × mean response
time
 The law applies as long as the number of jobs entering the
system is equal to completing service, so that no new jobs
are created in the system and no jobs are lost inside the
system.
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 The law can be applied to the part of the system
consisting of the waiting and serving positions because
once a job finds a waiting position , it is not lost.
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1. Discrete-State and Continuous-State Processes: A
process is called discrete or continuous state depending
upon the values its state can take.
 If the number of possible values is finite or countable, the
process is called a discrete-state process.
 The waiting time w(t) can take any value on the real line
then, w(t) is a continuous-state process.
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2. Markov Process: If the future states of a process are
independent of the past and depend only on the present,
the process is called a Markov process.
 It can be used to model a random system that change the
state according to transition rule that only depend on
current state.
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3. Birth-Death Processes: The discrete-space Markov
processes in which the transitions are restricted to
neighboring states only are called birth-death processes.
 It is possible to represent states by integer such that a
process in state n can change only two state n+1 or n-1.
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4. Poisson Processes: It is a random process which
counts the number of event and time that these event
occur in given time interval (t, t+x) has a Poisson
distribution, and then, the arrival process is referred to
as a Poisson process.
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Poisson process have the following properties:
 Merging of k Poisson process with mean rate λi results
in a Poisson stream with mean rate λ given by
λ =
 If a Poisson process is split into k substreams such that
the probability of a job going to the ith substream is pi,
each substream is also Poisson with a mean rate of piλ.
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
k
i
i
1


 If the arrivals to a single server with exponential service
time are Poisson with mean rate λ, the departures are also
Poisson with the same rate λ, provided the arrival rate λ is
less than the service rate μ.
 If the arrivals to a service facility with m service centres
are Poisson with a mean rate λ, then the departures also
a Poisson with the same rate λ and provided the arrival
rate λ is less than the total service rate
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Queuing theory is major for computer system
performance and also in our daily life. It can be used to
help reduce waiting times. It helps in determining the
time that the jobs spend in various queues in the system.
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Queuing theory

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