Queueing Theory and Models1

Operations Research TAPMI, Theme 3, B2008, Feb-Apr 2009 Prof. Ajith Kumar

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Course facilitators: Prof. Ajith Kumar

Prof. T N Badri

QUEUEING THEORY & MODELS

OPERATIONS RESEARCH


OORQUEUEING THEORY

The motivation to study queueing systems

In the US alone, people collectively spend 37,000,000,000 man-hours per year waiting in queues (Hillier and Lieberman, 2005)

This amounts to an opportunity loss of nearly 1.5 billion man-days of work per year! (this is for the US alone).

FROM THE PERSPECTIVE OF A COUNTRY


OORQUEUEING THEORYThe motivation to study queueing systems

Increasing service counters decreases customer waiting time and increases customer satisfaction. But costs of providing servicegoes up.

Decreasing service counters increases customer waiting time and decreases customer satisfaction. It decreases cost of providingservice, but cost of customer dissatisfaction goes up.

SO, WHAT IS THE OPTIMAL SERVICE LEVEL?

Total cost = Cost of providing service + Cost of customer dissatisfaction

If you reduce one of the components, the other one increases….

FROM THE PERSPECTIVE OF A COMPANY


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MODEL OF A QUEUEING

SYSTEMInput Characteristics

Output Characteristics

QUEUEING THEORY

The basic queueing model


OORThe inside of a queueing system

QUEUEING THEORY

S

customer receiving service

customers waiting (in the Q)

Waiting line (Q)server

Calling population

customer rejoins the population

customer does not

rejoin population

(Not in the Q)

A single-channel, single-phase system


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SSingle channel, Single phase

Single channel, Multi phase

Multi channel, Single phase

Multi channel, Multi phase

S1

S2

S3

S1S2S3

S1

S2

S3

T1

T2

T3

U

QUEUEING SYSTEMS INPUTS


OORInputs


Arrival rate, arrival time, and arrival size.

Type of population

Service characteristics (service rate, service time).

Exit fates (rejoins population, does not rejoin).

Capacities (of the systems, of the sub-systems).

Q behavior (balking, reneging, jockeying)

These are the typical inputs considered -

Each of these is explored briefly in the next few slides…

Q discipline (FIFO, LIFO, SIRO, SPT, PR)


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No. of customers that arrive in a given period of time.

ARRIVAL RATE

Given period is chosen as convenient (per minute; per hour; per day).Usually a variable, e.g. # customers arriving at a bank can be different between different 1-hour periods.

Often seen to follow the Poisson Process, and is modeled thus.

P(r) = (λre-λ) / r!r: number of arrivals in the given time period. r = 0, 1, 2, …. ∞

P(r): the probability that exactly n arrivals will occur in the given period.λ: mean arrival rate (mean number of arrivals in a given time period).

Inputs > Arrival Characteristics


But, in general, arrival rate can follow any distribution.


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The time elapsed between two consecutive arrivals into the system.

INTER-ARRIVAL TIME (t)

Usually a variable, e.g. the time between two customers arriving at a bank can be different

If the arrival behaviour follows a Poisson Process, then t follows the exponential distribution, and is modeled thus.

f(t) = λe-λt

f(t): the probability density function of t

λ: mean arrival rate = 1 / (mean of t).

Inputs > Arrival Characteristics


ARRIVAL SIZE Can be single arrivals, or batch arrivals


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No. of customers that a server can handle in a given time period.

SERVICE RATE

For a given server, this can vary from one period to the next period. Hence, this is a random variable and has a distribution.

As with the arrival process, the service process is often found to be Poisson.

Multiple server system: distribution, its mean, variance can vary from server to server.

Pi(X) = (µiXe-µ) / X!

X: number of customers served in the given time period. X = 0, 1, 2, …. ∞

Pi(X): the probability that exactly X customers will be served by server i in the given period.

µi: mean service rate (the mean of X) of server i.

Inputs > Service Characteristics

i = 0, 1, 2, …. m

m: number of servers (channels).



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Time taken by a server to handle a customer. For a given server, this can vary from customer to customer.

SERVICE TIME

Distribution can vary across servers.

Multiple server system: distribution, its mean, variance can vary from server to server.

Inputs > Service Characteristics

fi(t) = µie-µit

fi(t): the probability density function of t

µi: mean service rate at server i = (mean service time or server i)-1.



OORInputs > Population

Limited pool of customers.FINITE POPULATION

Type of the population influences the arrival characteristics of the system.

When customer leaves population, probability of next occurrence (or arrival rate) decreases, and vice versa.E.g. a small set of machines that need maintenance & service.

Unlimited pool of customers.INFINITE POPULATION

Customers leaving and rejoining the population does not influence arrival rate.

E.g. the customer pool of a large bank, or a big retail store.



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a part of the system may be able to hold only up to a certain number of customers.

CAPACITIES

particularly relevant in multi-phase systems, with finite capacities.

Inputs > Capacities

e.g. the number of cars that can stand in line for drying after a car-wash may be 10, hence car-wash has to become idle when the drying waiting line reaches 10.



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FIFO: First In First Out (FCFS: First Come First Served)

Inputs > Q discipline

A set of rules that determine the order of service offered to the customers in a given waiting line

LIFO: Last In First Out

SIRO: Service in Random Order

SPT: Shortest Processing Time First

PR: Service according to priority

Find real-life examples of each of these…


For all problems in this course, we assume the FIFO discipline


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Balking A customer does not enter the Q when he perceives its too long, and/or he does not have enough time to wait and avail the service.

Inputs > Q behavior

Do customers always wait in the line patiently? No, they also exhibit these behaviors -

Reneging A customer waits in the Q for some time, but leaves when he sees it moving too slowly, and he does not have enough time.

Jockeying A customer moves from his current line to another, e.g. if he feels he will receive service earlier by changing lines.



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CUSTOMER REJOINS CALLING POPULATION

Inputs > Exit Fates

A serviced machine that may need the same service again.

CUSTOMER DOES NOT REJOIN CALLING POPULATION

A cardiac patient passes away in a hospital.The machine overhauled, modified or repaired in such a way that there is a low probability of needing the same service again, or it is discarded.

Significantly influences the arrival characteristics of the system, when calling population is finite, but not when it is infinite.



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TOGETHER, THE INPUT CHARACTERISTICS HELP DEFINE THE

STRUCTURE OF THE QUEUEING SYSTEM


OORKENDALL QUEUEING NOTATION

A: distribution of inter-arrival time.if exponential/markov, M.if constant/deterministic, D.if Erlang or order k, Ekif phase-type, P Hif hyper-exponential, Hif arbitrary, or general, GIf general independent, G I

The Kendall notation has been widely adopted in queueing theory to depict the structure of the system in a simple manner.

A / B / c / N / K

B: distribution of service-time.(same notation as for A)

c: no. of channels / parallel servers.

N: System capacity

K: Size of the calling population.



Example 1: M / M / 1 / ∞ / ∞Single-channel (server) system;Both inter-arrival and service times follow exponential distributions;System capacity is unlimited; population is infinite

Example 2: G / G / 2 / 10 / 52 server system;Both inter-arrival and service times follow arbitrary distributions;System capacity is 10 customers; population has only 5 members

e.g. a set of 5 machines in a factory, maintained by 2 technicians, who can actually handle 10 machines with the resources available.



Simplified Kendall Notation

When system capacity and calling population are infinite, they can be excluded from the notation.

e.g. M / M / 1 / ∞ / ∞ can be written as M / M / 1

Many queueing systems can be approximated as having infinite calling population and system capacity; hence the simplified notation is popular.


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NEXT, THE OUTPUTS


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Server utilization, or the proportion of time the server is busy (ρ).

Long-run average time spent by a customer in the system (W) and the queue (Wq).

Long-run time-average of number of customers in the system (L) and the queue (Lq).

These – not the inputs – influence customer satisfaction / delight.

Q stability vs. instability.

QUEUEING SYSTEMS OUTPUTS

Outputs: These are our performance measures.

Long-run proportion of time, the system contains more than k customers.

Server idle-time, or the proportion of time the no customer is in the system(Po).


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STANDARD QUEUEING MODELS


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Service times vary from one customer to next & are independent; but their average rate is known.

Queue discipline: arrivals are served FIFO.

Important Assumptions

Average service rate (µ) is > average arrival rate (λ).


M/M/1: single-channel; exponential inter-arrival times & service times; infinite capacity & population

Queue behavior: no balking/reneging; every arrival waits & receives service.

Arrivals independent of each other; avg number of arrivals is constant over time.

Arrival rate follows a Poisson distribution; and come from an infinite population.

1


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λ = Mean # arrivals in given time period, (or mean arrival rate)

formulas for the performance measures

STANDARD QUEUEING MODELSM/M/1:

1. Avg. # customers in the system over time (L):

1

µ = Mean # customers served in given time period, (or mean service rate)

2. Avg. # customers in the queue over time (Lq):

3. Avg. time a customer spends in the system (W):

4. Avg. time a customer spends in the queue (Wq):

5. Utilization factor (ρ):

6. % idle time; probability that no one in the system (Po):

7. Probability that no. of customers greater than k (Pn > k):

L = λ / (µ – λ)

Lq = λ2 / µ(µ – λ)

W = 1 / (µ – λ)

Wq = λ / µ (µ – λ)

ρ = λ / µ

P0 = 1 – (λ / µ)

Pn > k = (λ / µ)k+1


OORSTANDARD QUEUEING MODELS2 M/M/m: multi-channel; exponential inter-arrival times &

service times; infinite capacity & population

Same as for a single-channel queueing system.

Important Assumptions


OORSTANDARD QUEUEING MODELS

M/M/m:2

Lq = L – λ/µ


L = P0 [ λ µ (λ/µ)m / (m-1)! (mµ – λ)2 ] + λ/µ

m: number of channels (servers)




W = L / λ

Wq = W – 1 / λ = Lq / λ

contd…

formulas for the performance measures


OORformulas for the performance measures


M/M/m:2

= λ / mµ

m: number of channels (servers)

5. Utilization factor (ρ):


1

Σ (1/n!) (λ/µ)n + (1/m!) (λ/µ)m (mµ/(mµ – λ)n = 0

n = m-1for mµ > λ

…contd




3




L = Lq + (λ / µ)

Lq = λ2 / 2µ(µ – λ)

W = Wq + 1 / (µ – λ)

Wq = λ / 2µ(µ – λ)

M/D/1: single-channel; exponential inter-arrival times, constant service times; infinite capacity & population.




4




L = Lq + (1 – P0)

Lq = N – [(λ + µ) / λ ] (1 – P0)

W = Wq + 1 / µ

Wq = Lq / λ(K – L)

M / M / 1 / ∞ / K: single-channel, exponential inter-arrival time and service times, finite population of size K


1

Σ [K!/(K-n)!] (λ/µ)nn = 0

n = K

6. Probability of n units in the system (Pn): P0 [K!/(K-n)!] (λ/µ)n



General Relationships under Steady State Conditions

Applicable for any queue, under steady state (except the finite population model)

(or W = L / λ)L = λ W

(or Wq = Lq / λ)Lq = λ Wq

Little’s Flow Equations

Also, avg. time in system = avg. time in q + avg. time in service

W = Wq + 1 / µ

The formulas are applicable under steady-state conditions.


OORThe preceding formulas are applicable under steady-state conditions, and under the respective assumptions made.

What should we do when one or more of the assumptions do not hold and/or the queue in a transient state?

SIMULATION MODELS ARE DEVELOPED FOR THIS

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Queueing Theory and Models1