Queueing Models. A Basic Queueing System Herr Cutter’s Barber Shop Herr Cutter is a German barber who runs a one-man barber shop. Herr Cutter opens his.

A Basic Queueing System

CustomersQueue

Served Customers

Queueing System

Service facility

SSSS

CCCC

C C C C C C C

Served Customers

Herr Cutter’s Barber Shop

• Herr Cutter is a German barber who runs a one-man barber shop.

• Herr Cutter opens his shop at 8:00 A.M.

• The table shows his queueing system in action over a typical morning.

CustomerTime ofArrival

HaicutBegins

Durationof Haircut

HaircutEnds

1 8:03 8:03 17 minutes 8:20

2 8:15 8:20 21 minutes 8:41

3 8:25 8:41 19 minutes 9:00

4 8:30 9:00 15 minutes 9:15

5 9:05 9:15 20 minutes 9:35

6 9:43 — — —

Arrivals

• The time between consecutive arrivals to a queueing system are called the interarrival times.

• The expected number of arrivals per unit time is referred to as the mean arrival rate.

• The symbol used for the mean arrival rate is

= Mean arrival rate for customers coming to the queueing system

where is the Greek letter lambda.

• The mean of the probability distribution of interarrival times is

1 / = Expected interarrival time

• Most queueing models assume that the form of the probability distribution of interarrival times is a Poisson and Service an Exponential distribution

The Queue

• The number of customers in the queue (or queue size) is the number of customers waiting for service to begin.

• The number of customers in the system is the number in the queue plus the number currently being served.

• The queue capacity is the maximum number of customers that can be held in the queue.

• An infinite queue is one in which, for all practical purposes, an unlimited number of customers can be held there.

• When the capacity is small enough that it needs to be taken into account, then the queue is called a finite queue.

• The queue discipline refers to the order in which members of the queue are selected to begin service.

– The most common is first-come, first-served (FCFS).– Other possibilities include random selection, some priority procedure, or even last-

come, first-served.

Service

• When a customer enters service, the elapsed time from the beginning to the end of the service is referred to as the service time.

• Basic queueing models assume that the service time has a particular probability distribution.

• The symbol used for the mean of the service time distribution is

1 / = Expected service time

where is the Greek letter mu.

• The interpretation of itself is the mean service rate.

= Expected service completions per unit time for a single busy server

Labels for Queueing Models

To identify which probability distribution is being assumed for service times (and for interarrival times), a queueing model conventionally is labeled as follows:

Distribution of service times

— / — / — Number of Servers

Distribution of interarrival times

The symbols used for the possible distributions areM = Exponential distribution (Markovian)D = Degenerate distribution (constant times)Ek = Erlang distribution (shape parameter = k)GI = General independent interarrival-time distribution (any distribution)G = General service-time distribution (any arbitrary distribution)

Summary of Usual Model Assumptions

1. Interarrival times are independent and identically distributed according to a specified probability distribution.

2. All arriving customers enter the queueing system and remain there until service has been completed.

3. The queueing system has a single infinite queue, so that the queue will hold an unlimited number of customers (for all practical purposes).

4. The queue discipline is first-come, first-served.

5. The queueing system has a specified number of servers, where each server is capable of serving any of the customers.

6. Each customer is served individually by any one of the servers.

7. Service times are independent and identically distributed according to a specified probability distribution.

Examples of Commercial Service SystemsThat Are Queueing Systems

Type of System Customers Server(s)

Barber shop People Barber

Bank teller services People Teller

ATM machine service People ATM machine

Checkout at a store People Checkout clerk

Plumbing services Clogged pipes Plumber

Ticket window at a movie theater People Cashier

Check-in counter at an airport People Airline agent

Brokerage service People Stock broker

Gas station Cars Pump

Call center for ordering goods People Telephone agent

Call center for technical assistance People Technical representative

Travel agency People Travel agent

Automobile repair shop Car owners Mechanic

Vending services People Vending machine

Dental services People Dentist

Roofing Services Roofs Roofer

Examples of Internal Service SystemsThat Are Queueing Systems

Type of System Customers Server(s)

Secretarial services Employees Secretary

Copying services Employees Copy machine

Computer programming services Employees Programmer

Mainframe computer Employees Computer

First-aid center Employees Nurse

Faxing services Employees Fax machine

Materials-handling system Loads Materials-handling unit

Maintenance system Machines Repair crew

Inspection station Items Inspector

Production system Jobs Machine

Semiautomatic machines Machines Operator

Tool crib Machine operators Clerk

Examples of Transportation Service SystemsThat Are Queueing Systems

Type of System Customers Server(s)

Highway tollbooth Cars Cashier

Truck loading dock Trucks Loading crew

Port unloading area Ships Unloading crew

Airplanes waiting to take off Airplanes Runway

Airplanes waiting to land Airplanes Runway

Airline service People Airplane

Taxicab service People Taxicab

Elevator service People Elevator

Fire department Fires Fire truck

Parking lot Cars Parking space

Ambulance service People Ambulance

Defining the Measures of Performance

L = Expected number of customers in the system, including those being served (the symbol L comes from Line Length).

Lq = Expected number of customers in the queue, which excludes customers being served.

W = Expected waiting time in the system (including service time) for an individual customer (the symbol W comes from Waiting time).

Wq = Expected waiting time in the queue (excludes service time) for an individual customer.

These definitions assume that the queueing system is in a steady-state condition.

Relationship between L, W, Lq, and Wq

• Since 1/ is the expected service time

W = Wq + 1/

• Little’s formula states that

L = W

and

Lq = Wq

• Combining the above relationships leads to

L = Lq +

Using Probabilities as Measures of Performance

• In addition to knowing what happens on the average, we may also be interested in worst-case scenarios.

– What will be the maximum number of customers in the system? (Exceeded no more than, say, 5% of the time.)

– What will be the maximum waiting time of customers in the system? (Exceeded no more than, say, 5% of the time.)

• Statistics that are helpful to answer these types of questions are available for some queueing systems:

– Pn = Steady-state probability of having exactly n customers in the system.

– P(W ≤ t) = Probability the time spent in the system will be no more than t.

– P(Wq ≤ t) = Probability the wait time will be no more than t.

• Examples of common goals:– No more than three customers 95% of the time: P0 + P1 + P2 + P3 ≥ 0.95

– No more than 5% of customers wait more than 2 hours: P(W ≤ 2 hours) ≥ 0.95

Notation for Single-Server Queueing Models

• = Mean arrival rate for customers= Expected number of arrivals per unit time

1/ = expected interarrival time

• = Mean service rate (for a continuously busy server)= Expected number of service completions per unit time

= expected service time

• = the utilization factor= the average fraction of time that a server is busy serving customers=

The M/M/1 Model

• Assumptions1. Interarrival times have an exponential distribution with a mean of 1/.

2. Service times have an exponential distribution with a mean of 1/.

3. The queueing system has one server.

• The expected number of customers in the system is

L = 1 – = –

• The expected waiting time in the system is

W = (1 / )L = 1 / ( – )

• The expected waiting time in the queue is

Wq = W – 1/ = / [( – )]

• The expected number of customers in the queue is

Lq = Wq = 2 / [( – )] = 2 / (1 – )

The M/M/1 Model

• The probability of having exactly n customers in the system is

Pn = (1 – )n

Thus,P0 = 1 – P1 = (1 – )P2 = (1 – )2

::

• The probability that the waiting time in the system exceeds t is

P(W > t) = e–(1–)t for t ≥ 0

• The probability that the waiting time in the queue exceeds t is

P(Wq > t) = e–(1–)t for t ≥ 0