Introduction to queueing theory

Indian Institute of Technology Kharagpur

Queuing Theory

Prepared by:Pranav Mishra



Queuing Theory

•Queuing theory is the mathematics of waiting lines.

•A queue forms whenever existing demand exceeds the existing capacity of

service facility.

•It is extremely useful in predicting and evaluating system performance.

Queuing system

Key elements of queuing systema) Customers : Refers to anything that arrives at facility and

requires serviceb) Servers: Refers to any resource that provides the

requested service

inputServer

Queue

output Customer


Applications of queuing theory•Telecommunications

•Traffic control

•Layout of manufacturing systems

•Airport traffic

•Ticket sales counter, etc.

Examples: System Customers ServerReception desk People Receptionist

Hospital Patients Nurses

Airport Airplanes Runway

Road network Cars Traffic light

Grocery Shoppers Checkout station

Computer Jobs CPU, disk, CD


Components of a queuing system process

Dave’s Car Wash

enter exit

Population ofdirty cars

Arrivalsfrom thegeneralpopulation …

Queue(waiting line)

Servicefacility

Exit the system

Exit the systemArrivals to the system In the system

1) Arrival process2) Queue configuration

3) Queue discipline

4) Service discipline

5) Service facility



1) Arrival process•The source may be,

•single or multiple.

•Size of the population may be,•finite or infinite.

•Arrival may be single or bulk.•Control on arrival may be,

•Total control.•Partial control.•No control

•Statistical distribution of arrivals may be,•Deterministic,•Probabilistic.



2) Queue configuration

•The queue configuration refers to,• number of queues in the system,•Their spatial consideration,•Their relationship with server.

•A queue may be single or multiple queue.•A queuing system may impose restriction on the maximum number of units allowed.

3) Queue discipline

•If the system is filled to capacity, arriving unit is not allowed to join the queue.•Balking – A customer does not join the queue.•Reneging – A customer joins the queue and subsequently decides to leave.•Collusion – Customers collaborate to reduce the waiting time.•Jockeying – A customer switching between multiple queues.•Cycling – A customer returning to the queue after being served.

•A queue may be single or multiple queue.



4) Service discipline

•First In First Out (FIFO) a.k.a First Come First Serve (FCFS)•Last In First Out (LIFO) a.k.a Last Come First Served (LCFS).•Service In Random Order (SIRO).•Priority Service

•Preemptive•Non-preemptive

5) Service facility

•Single queue single server

Queue

Arrivals Service facility

Departuresafter service



5) Service facility

•Single queue multiple server

Queue

Service facility

Channel 1

Service facility

Channel 2

Service facility

Channel 3

Arrivals Departuresafter service



5) Service facility

•Multiple queue multiple server

Service stationCustomers leaveQueuesArrivals



5) Service facility

•Multiple server in series

Arrivals

Queues

Service station 1 Service station 2

QueuesCustomers leave

Phase 1 Phase 2


Queuing models : some basic relationships

= Mean number of arrivals per time period

µ = Mean number of units served per time period

Assumptions:

• If > µ, then waiting line shall be formed and increased indefinitely and

service system would fail ultimately.

• If < µ, there shall be no waiting line.

Average number of units (customers) in the system (waiting and being served)

= / (µ - )

Average time a unit spends in the system (waiting time plus service time)

= 1/ (µ - )


Queuing models : some basic relationships

Average number of units waiting in the queue

= 2/ µ(µ - )

Average time a unit spends waiting in the queue

= / µ(µ - )

Intensity or utilization factor

= / µ


Special Delay studies

a) Merging delays

b) Peak flow delay

c) Parking

a) Merging delays

Merging may be defined as absorption of one group of traffic by another.

Oliver & Bisbee postulated that minor stream queue length are function of

major street flow rates.

This model assumes that:

•A gap of at-least T is required to enter the major stream.

•Only one entry is permitted through one acceptable gap.

•Entries occur just after passing of vehicles, that signals beginning of gap of

acceptable size.



• Appearance of gap in major stream is not affected by queue in minor

stream; and,

• Arrivals into the minor stream queue are Poisson

Average number of vehicles in minor stream queue E(n)

Where,

qa = minor stream flow,

qb = major stream flow,

T = minimum acceptable gap



This model works particularly better for the situation,

• Where major stream flow rate is high, and

• Vehicles in minor stream queue are served on FIFO basis, with the

appearance of a minimum acceptable gap T.

HO formulated a model to predict the amount of time required to clear two

joining traffic streams through a merging point.

This model assumes that:

•Merging is permitted only at merging point.

•Vehicles are served in FIFO basis.



The total time required for n1 and n2 vehicles to pass through the merging

point is,

Where,

hi = i’th time gap on major road,

to = time required for a vehicle to merge into through traffic, assuming all

vehicles take same time to merge.

α = number of vehicle that merge into i’th gap.

n1 = number of vehicle in major road,

n2 = number of vehicle waiting to merge



• If traffic demand exceeds the capacity, there is a continuous buildup of

traffic.

• Mean service rate exceeds the mean rate of arrival.

• Expected number of vehicle ‘n’, waiting in the system at any time ‘t’ can be

represented as E[n(t)] and will grow indefinitely as ‘t’ increases.

b) Peak flow delay

E[n(t)] = E(n) + λ(t) - μ(t)

• E(n) = expected number of vehicles in system with initial traffic intensity ρo,

where, ρo <1

• λ = mean arrival rate and, μ = mean service rate



• Now, say traffic intensity ρo increases to ρ1 , where ρ1 >1

• Therefore, ρ1 = λ / μ [ initial λ0 increases to λ]

or λ = μ . ρ1

• So, E[n(t)] = E(n) - μ . ρ1 (t) - μ(t)

E[n(t)] = E(n) + (ρ1 - 1) μ . t

Or, E[n(t)] = ρ0 /(1 - ρ0 ) + (ρ1 - 1) μ . t

• When, service rate (μ) is constant,

E[n(t)] = (1/2) λ0 2 / μ(μ - λ0 ) + λ0/μ + (ρ1 - 1) μ



Numerical example:

• A queue with random arrival rate 1 vehicle per minute and a mean service

time of 45 seconds. In peak period, arrival rate suddenly doubles and this

peak period rate is maintained for 1 hour. Find the average number of

vehicles in the system at the end of peak hour.

Sol. – Given, λ0 = 1, μ = 4/3 Therefore, ρ0 = λ0 / μ = 3/4

In peak period, λ = 2 and μ remains same. So, ρ1 = 3/2

Putting the values in eqn - , E[n(t)] = ρ0 /(1 - ρ0 ) + (ρ1 - 1) μ . t

we get, E[n(60)] = 43

If the service rate μ were constant,

Putting the values in eqn - , E[n(t)] = (1/2) λ0 2 / μ(μ - λ0 ) + λ0/μ + (ρ1 - 1) μ

we get, E[n(60)] = 41.87~ 42



Now, to find out how long it takes the peak hour queue to dissipate, COX

developed the equation =

For developing this model, he made following assumption:

• Service time is constant.

• When traffic starts to dissipate, there are large number of vehicles in the

queue and traffic intensity ρ1 has decreased to less than 1.

• The queuing time of newly arrived vehicle is equal to average queuing time

of vehicles already in the system.



• For the previous problem, find out the mean time it takes for queue to get

dissipated.

Sol: putting the values in the equation

E (t) = [ E(n)t /μ – ρo / 2(1- ρo ) ] / (1 – ρo )

We get, E(t) = 123 min.



• The characteristics of queuing analysis dealing with length of queue and

waiting time are not too meaningful for parking as potential parkers usually

leave and seek another location rather than wait, if parking is full

• Though there has been attempts to establish relationship between number

of potential parkers turned away from parking of a specified capacity and

various fractions of occupancy.

c) Parking

Introduction to queueing theory

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