1. Queueing Models

2. A Basic Queueing System
Figure A basic queueing system, where each customer is indicated by C and each server by S. Although this figure shows four servers, some queueing systems (including the example in this section) have only a single server.

3. 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.
Customer
Time of Arrival
Haicut Begins
Duration of Haircut
Haircut Ends 1
8:03
17 minutes
8:20 2
8:15
21 minutes
8:41 3
8:25
19 minutes
9:00 4
8:30
15 minutes
9:15 5
9:05
20 minutes
9:35 6
9:43 —
Table The data for Herr Cutter’s first five customers.

4. 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
l = Mean arrival rate for customers coming to the queueing system
where l is the Greek letter lambda.
The mean of the probability distribution of interarrival times is
1 / l = 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

5. 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.

6. 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 / m = Expected service time where m is the Greek letter mu.
The interpretation of m itself is the mean service rate. m = Expected service completions per unit time for a single busy server

7. 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 are M = 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)

8. Summary of Usual Model Assumptions
Interarrival times are independent and identically distributed according to a specified probability distribution.
All arriving customers enter the queueing system and remain there until service has been completed.
The queueing system has a single infinite queue, so that the queue will hold an unlimited number of customers (for all practical purposes).
The queue discipline is first-come, first-served.
The queueing system has a specified number of servers, where each server is capable of serving any of the customers.
Each customer is served individually by any one of the servers.
Service times are independent and identically distributed according to a specified probability distribution.

9. Examples of Commercial Service Systems That Are Queueing Systems
Type of System
Customers
Server(s)
Barber shop
People
Barber
Bank teller services
Teller
ATM machine service
ATM machine
Checkout at a store
Checkout clerk
Plumbing services
Clogged pipes
Plumber
Ticket window at a movie theater
Cashier
Check-in counter at an airport
Airline agent
Brokerage service
Stock broker
Gas station
Cars
Pump
Call center for ordering goods
Telephone agent
Call center for technical assistance
Technical representative
Travel agency
Travel agent
Automobile repair shop
Car owners
Mechanic
Vending services
Vending machine
Dental services
Dentist
Roofing Services
Roofs
Roofer
Table Examples of commercial service systems that are queueing systems.

10. Examples of Internal Service Systems That Are Queueing Systems
Type of System
Customers
Server(s)
Secretarial services
Employees
Secretary
Copying services
Copy machine
Computer programming services
Programmer
Mainframe computer
Computer
First-aid center
Nurse
Faxing services
Fax machine
Materials-handling system
Loads
Materials-handling unit
Maintenance system
Machines
Repair crew
Inspection station
Items
Inspector
Production system
Jobs
Machine
Semiautomatic machines
Operator
Tool crib
Machine operators
Clerk
Table Examples of internal service systems that are queueing systems.

11. Examples of Transportation Service Systems That 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
Airline service
People
Airplane
Taxicab service
Taxicab
Elevator service
Elevator
Fire department
Fires
Fire truck
Parking lot
Parking space
Ambulance service
Ambulance
Table Examples of transportation service systems that are queueing systems.

12. 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.

13. Relationship between L, W, Lq, and Wq
Since 1/m is the expected service time W = Wq + 1/m
Little’s formula states that L = lW and Lq = lWq
Combining the above relationships leads to L = Lq + l/m

14. 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

15. Notation for Single-Server Queueing Models
l = Mean arrival rate for customers = Expected number of arrivals per unit time 1/l = expected interarrival time
m = Mean service rate (for a continuously busy server) = Expected number of service completions per unit time 1/m = expected service time
r = the utilization factor = the average fraction of time that a server is busy serving customers = l / m

16. The M/M/1 Model Assumptions
Interarrival times have an exponential distribution with a mean of 1/l.
Service times have an exponential distribution with a mean of 1/m.
The queueing system has one server.
The expected number of customers in the system is
L = r / (1 – r) = l / (m – l)
The expected waiting time in the system is
W = (1 / l)L = 1 / (m – l)
The expected waiting time in the queue is
Wq = W – 1/m = l / [m(m – l)]
The expected number of customers in the queue is
Lq = lWq = l2 / [m(m – l)] = r2 / (1 – r)

17. The M/M/1 Model
The probability of having exactly n customers in the system is
Pn = (1 – r)rn Thus, P0 = 1 – r P1 = (1 – r)r P2 = (1 – r)r2 : :
The probability that the waiting time in the system exceeds t is
P(W > t) = e–m(1–r)t for t ≥ 0
The probability that the waiting time in the queue exceeds t is
P(Wq > t) = re–m(1–r)t for t ≥ 0

18. Why is There Waiting? Example #1: McDonalds
50 customers arrive per hour
Service rate is 60 customers per hour
Example #2: Doctor’s Office
Arrivals are scheduled to arrive every 20 minutes.
The doctor spends an average of 18 minutes with each patient.
For each example, ask if there will be waiting.
For example #1, they can serve customers more quickly than they arrive. Will there still by waiting? Yes, because of variability in arrivals. Arrivals tend to come in bunches (e.g., a bus full of little-leaguers after a game, or a whole large family) causing lines to form. Also, 50 customers per hour may be an average. During the busy lunch hour, the arrival rate may be higher, and lines will form.
For example #2, there is little to no variability in arrivals. Will there still be waiting? Yes, because of variability in service. A really long service (i.e., for a very sick patient) will cause the following appointments to be delayed. The doctor being called away on an emergency will cause the appointments to be delayed.

19. System Characteristics
Number of servers
Arrival and service pattern
rate of arrivals and service
distribution of arrivals and service
Maximum size of the queue
Queue disciplince
FCFS?
Priority system?
Population size
Infinite or finite?

20. Measures of System Performance
Average number of customers waiting
in the system
in the queue
Average time customers wait
Which measure is the most important?
When customers are internal to the organization, the first measure tends to be more important: Having such customers wait causes lost productivity.
Commercial service systems tend to place greater importance on the second measure: Outside customers are typically more concerned with how long they have to wait than with how many customers are there.

21. Number of Servers Single Server Multiple Servers
Discuss examples of each:
Single Server -- Wendys
Multiple server (one line): bank, airport
Multiple server (line for each server): McDonalds, grocery store
These two multiple-server setups are identical from a modeling perspective. The average line lengths and waiting times will be identical. However, the left one is more fair, because it always is FCFS. In the right one, a customer can get “unlucky” and get in a slow line, and be served after someone who arrived after them but got in a quicker line. Studies have shown that waiting does not bother customers so much as perceived unfairness does.
Something else to discuss: What’s better, a single-server with many people working together to make the service go fast (the model at Wendys), vs. multiple servers, with one person at each that does the whole service more slowly (the model at McDonalds). We will return to examine this question in the following lecture.

22. Arrival Pattern A Poisson distribution is usually assumed.
A good approximation of random arrivals.
Lack-of-memory property: Probability of an arrival in the next instant is constant, regardless of the past.

23. Service Pattern Either an exponential distribution is assumed,
Implies that the service is usually short, but occasionally long
If service time is exponential then service rate is Poisson
Lack-of-memory property: The probability that a service ends in the next instant is constant (regardless of how long its already gone).
Decent approximation if the jobs to be done are random.
Not a good approximation if the jobs to be done are always the same.
Or any distribution
Only single-server model is easily solved.

24. Maximum Size of Queue
Most queueing models assume an infinite queue length is possible.
If the queue length is limited, a finite queue model can be used.
Examples with a finite queue: call center (telephone lines limited), drive through lane.

25. Queue Discipline
Most queueing systems assume customers are served first-come first-served.
If certain customers are given priority, a priority queueing model can be used.
Nonpreemptive: Finish customer in service before taking a new one.
Preemptive: If priority customer arrives, any regular customer in service is preempted (put back in the queue).

26. Population Source
Most queueing models assume an infinite population source.
If the number of potential customers is small, a finite source model can be used.
Number in system affects arrival rate (fewer potential arrivals when more in system)
Okay to assume infinite if N > 20.
Example with a finite source: Equipment awaiting repair.

27. Models Single server, exponential service time (M/M/1)
Single server, general service time (M/G/1)
Multiple servers, exponential service time (M/M/s)
Finite queue (M/M/s/K)
Priority queue (nonpreemptive and preemptive)
Finite calling population
A Taxonomy
— / — / — (and an optional fourth element / —)
Arrival Service Number of Maximum Distribution Distribution Servers in Queue where M = Exponential (Markovian) D = deterministic (constant) G = general distribution
Note that models 4 and 6 are not contained in the main text, but are included on the CD supplement for Chapter 14.

28. Notation
Parameters:
l = customer arrival rate m = service rate (1/m = average service time) s = number of servers
Performance Measures
Lq = average number of customers in the queue L = average number of customers in the system Wq = average waiting time in the queue W = average waiting time (including service) Pn = probability of having n customers in the system r = system utilization
Arrival rate and service rate must be in the form of a rate (customers per unit time) and must be in the same units.