queuing theory & models 1

Demand for a service facility >

Capacity of the facility.OR

Number of service facilities >

Number of customers requiring service.

Collection of mathematical models of various queuing systems.

Used to analyze production and service processes exhibiting random variability in market demand (arrival times) and service times.

Calling Population Jobs

Service Mechanism

Queue

Input Source Service System

Served Jobs

Leave the system

1.Arrival Process 2.Queue

Configuration

3. Queue Discipline

4.Service process

An input source is characterized by Size of the calling population :

Source – Finite OR Infinite Time – Known Schedule OR randomly Number- Individuals OR Groups

Behavior of arrivals: Patient OR impatient

2 aspects of a service system: configuration of the service system

speed of the service• Service time : amount of time needed to

service a customer. • Service Rate : number of customers

serviced during a particular time period.

Single Server Multiple Server

Finite Queue LengthInfinite Queue Length

A. Single Server – Single Queue

Example: Students arriving at a library counter

Service System

ArrivalsQueue Customer

Leaves

B. Single Server – Several Queues

Example: Service Calls at police stations

Service System

Arrivals Queue Customer Leaves

C. Several Server – Single Queue

Example: Mc Donalds

Service System

Arrivals Queue Customer LeavesService System

Service System

D. Several Server – Several Queues

Example: Immigration Counters at the Airports,

Cash Counters at Shopping Malls

Service System

Arrivals QueueCustomer Leaves

Service System

Service System

Number of Queues layout of a service system Length of Queues physical space, legal restrictions, and attitude of the customers

Static Queue Disciplines FCFS - prepaid taxi queues at airport LCFS - people who join an elevator

Dynamic Queue Disciplines SIRO – equal probability of being

served Priority - on basis of attributes or

urgency

A Queuing Model is used to: approximate a real queuing situation or system.

Represented using Kendall’s notation

A / B / S / C/ N /P› A : distribution of inter arrival time › B : distribution of service time› S : number of parallel servers› C: System Capacity› N : Population Size› P : Queuing Discipline

M / M / 1 / ∞ / ∞ single server queue with Poisson arrivals and exponential service times, a.k.a. M/M/1

M / M / S / ∞ / ∞ S server queue with Poisson arrivals and exponential service times, a.k.a. M/M/S

M / M / 1 / C / ∞ same as M/M/1 with finite system capacity C

M / M / 1 / C / K single server queue with Poisson arrivals from a population of K potential customers;and exponential service times

5 people enter a store per hour and the store serves 4 people per hour. The stores is unable to serve its customers. Suggest the minimum number of people the store must serve to handle the rush

Arrival rate (λ) = 5/ hr Service rate (µ) = 4/ hr

Infinite queue length = λ/µ = 5/4 > 1

But the ratio should be less than 1 to serve adequately.

To make λ/µ < 1, µ can be increased. So the optimum service rate

should be 6 instead of 4. (µ = 6)

Used to model single processor systems or individual devices in a computer system.

The first part represents the input process, the second the service distribution, and the third the number of servers.

The M/M/1 Waiting line system has a Single channel Single phase Poisson arrival rate , λ Exponential service time, µ Unlimited population First-in First-out queue discipline

M/M/1 Operating Characteristics

λ , µ

Utilization(fraction of time server is busy)

ρ = λ / µ

Average waiting times W = 1/(µ - λ) Wq = ρ/(µ - λ) = ρ W

Average number waiting L = λ /(µ - λ) Lq = ρ λ /(µ - λ) = ρ L

On a network gateway, measurements show that the packets arrive at a mean rate of 125 packets per seconds(pps) and the gateway takes about 2 milliseconds to forward them.

Using an M/M/1 model, analyze the gateway.

What is the probability of buffer overflow if the gateway had only 13 buffers? How many buffers do we need to keep packet loss below one packet per million?

Arrival rate = 125pps

Service rate = 1/.002 = 500 pps

Gateway utilization ρ = / = 0.25

Probability of n packets in the gateway › (1- ρ) ρ n = 0.75(0.25)n

Mean time spent in the gateway› (1/ )/(1- ρ) = (1/500)/(1-0.25) = 2.66 milliseconds

Probability of buffer overflow› P(more than 13 packets in gateway) = ρ13 = 0.2313 =1.49 X 10-8 ≈ 15 packets

per billion packets

To limit the probability of loss to less than 1 packet per billion ( 10-6 )› ρ n < 10-6

› n > log(10-6)/log(0.25) = 9.96› Need about 10 buffers

Used for calculating waiting line characteristics when more than one server is involved

Two types› M/M/s finite queue› M/M/s finite (calling) population

It follows Kendal’s system, indicating:

Arrivals are a Poisson process Service time is exponentially distributed. There are ‘s’ number of servers The length of the arrival queue is infinite The population of users to be served by the

system is infinite

http://businessmanagementcourses.org/Lesson21QueuingTheory.pdf

http://www.stanford.edu/class/msande121/Handouts/qt.pdf

http://www.eventhelix.com/realtimemantra/congestioncontrol/queueing_theory.htm

http://staff.um.edu.mt/jskl1/simweb/mm1.htm