1. 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.

2. 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.

3. 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 = Mean interarrival time
Most queueing models assume that the form of the probability distribution of interarrival times is an exponential distribution.

4. Evolution of the Number of Customers
Figure The evolution of the number of customers in Herr Cutter’s barber shop over the first 100 minutes (from 8:00 to 9:40), given the data in Table 14.1.

5. The Exponential Distribution for Interarrival Times
Figure The shape of an exponential distribution, commonly used in queueing models as the distribution of interarrival times (and sometimes as the distribution of service times as well).

6. Properties of the Exponential Distribution
There is a high likelihood of small interarrival times, but a small chance of a very large interarrival time. This is characteristic of interarrival times in practice.
For most queueing systems, the servers have no control over when customers will arrive. Customers generally arrive randomly.
Having random arrivals means that interarrival times are completely unpredictable, in the sense that the chance of an arrival in the next minute is always just the same.
The only probability distribution with this property of random arrivals is the exponential distribution.
The fact that the probability of an arrival in the next minute is completely uninfluenced by when the last arrival occurred is called the lack-of-memory property.

7. 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.

8. 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 = Mean 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

9. Some Service-Time Distributions
Exponential Distribution
The most popular choice.
Much easier to analyze than any other.
Although it provides a good fit for interarrival times, this is much less true for service times.
Provides a better fit when the service provided is random than if it involves a fixed set of tasks.
Standard deviation: s = Mean
Constant Service Times
A better fit for systems that involve a fixed set of tasks.
Standard deviation: s = 0.
Erlang Distribution
Fills the middle ground between the exponential distribution and constant.
Has a shape parameter, k that determines the standard deviation.
In particular, s = mean / (k)

10. Standard Deviation and Mean for Distributions
Exponential
mean
Degenerate (constant)
Erlang, any k
(1 / k) (Mean)
Erlang, k = 2
(1 / 2) (Mean)
Erlang, k = 4
(1 / 2) (Mean)
Erlang, k = 8
(1 / 22) (Mean)
Erlang, k = 16
(1 / 4) (Mean)
Table The relationship between the standard deviation and the mean for service-time distributions

11. 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)

12. 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.

13. 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.

14. 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.

15. 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.

16. Choosing a Measure of Performance
Managers who oversee queueing systems are mainly concerned with two measures of performance:
How many customers typically are waiting in the queueing system?
How long do these customers typically have to wait?
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.

17. 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.

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

19. 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

20. The Dupit Corp. Problem
The Dupit Corporation is a longtime leader in the office photocopier marketplace.
Dupit’s service division is responsible for providing support to the customers by promptly repairing the machines when needed. This is done by the company’s service technical representatives, or tech reps.
Current policy: Each tech rep’s territory is assigned enough machines so that the tech rep will be active repairing machines (or traveling to the site) 75% of the time.
A repair call averages 2 hours, so this corresponds to 3 repair calls per day.
Machines average 50 workdays between repairs, so assign 150 machines per rep.
Proposed New Service Standard: The average waiting time before a tech rep begins the trip to the customer site should not exceed two hours.

21. Alternative Approaches to the Problem
Approach Suggested by John Phixitt: Modify the current policy by decreasing the percentage of time that tech reps are expected to be repairing machines.
Approach Suggested by the Vice President for Engineering: Provide new equipment to tech reps that would reduce the time required for repairs.
Approach Suggested by the Chief Financial Officer: Replace the current one-person tech rep territories by larger territories served by multiple tech reps.
Approach Suggested by the Vice President for Marketing: Give owners of the new printer-copier priority for receiving repairs over the company’s other customers.

22. The Queueing System for Each Tech Rep
The customers: The machines needing repair.
Customer arrivals: The calls to the tech rep requesting repairs.
The queue: The machines waiting for repair to begin at their sites.
The server: The tech rep.
Service time: The total time the tech rep is tied up with a machine, either traveling to the machine site or repairing the machine. (Thus, a machine is viewed as leaving the queue and entering service when the tech rep begins the trip to the machine site.)

23. 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

24. 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)]

25. 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) = r e–m (1–r)t for t ≥ 0

26. M/M/1 Queueing Model for the Dupit’s Current Policy
Figure This spreadsheet shows the results from applying the M/M/1 model with l = 3 and m = 4 to the Dupit case study under the current policy.

27. John Phixitt’s Approach (Reduce Machines/Rep)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).
John Phixitt’s suggested approach is to lower the tech rep’s utilization factor sufficiently to meet the new service requirement. Lower r = l / m, until Wq ≤ 1/4 day, where l = (Number of machines assigned to tech rep) / 50.

28. M/M/1 Model for John Phixitt’s Suggested Approach (Reduce Machines/Rep)
Figure This application of the spreadsheet in Figure 14.5 shows that, when m = 4, the M/M/1 model gives an expected waiting time to begin service of Wq = 0.25 days (the largest value that satisfies Dupit’s proposed new service standard) when l is changed from l = 3 to l = 2.

29. The M/G/1 Model Assumptions
Interarrival times have an exponential distribution with a mean of 1/l.
Service times (T) can have any probability distribution.
E(T) = 1/m , Var(T) = s2.
3. The queueing system has one server.
The probability of zero customers in the system is
P0 = 1 – r
The expected number of customers in the queue is
Lq = l2[Var(T)+ E(T)2] / [2(1 – lE(T))]
The expected number of customers in the system is
L = Lq + l/m
The expected waiting time in the queue is
Wq = Lq / l
The expected waiting time in the system is
W = Wq + 1/m

30. The Values of s and Lq for the M/G/1 Model with Various Service-Time Distributions
The expected number of customers in the queue is
Lq = l2[Var(T)+ E(T)2] / [2(1 – lE(T))]= [l2s2 + r2] / [2(1 – r)]
Distribution
Mean s
Model Lq
Exponential
1/m
M/M/1
r2 / (1 – r)
Degenerate (constant)
M/D/1
(1/2) [r2 / (1 – r)]
Erlang, with shape parameter k
(1/k) (1/m)
M/Ek/1
(k+1)/(2k) [r2 / (1 – r)]
Table The values of s and Lq for the M/G/1 model with various service-time distributions.

31. VP for Engineering Approach (New Equipment)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).
The Vice President for Engineering has suggested providing tech reps with new state-of-the-art equipment that would reduce the time required for the longer repairs.
After gathering more information, they estimate the new equipment would have the following effect on the service-time distribution:
Decrease the mean from 1/4 day to 1/5 day.
Decrease the standard deviation from 1/4 day to 1/10 day.

32. M/G/1 Model for the VP of Engineering Approach (New Equipment)
Figure This Excel template for the M/G/1 model shows the results from applying this model to the approach suggested by the Dupit’s vice president for engineering to use state-of-the-art equipment.

33. The M/M/s 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.
Any number of servers (denoted by s).
With multiple servers, the formula for the utilization factor becomes r = l / sm but still represents that average fraction of time that individual servers are busy.

34. Values of L for the M/M/s Model for Various Values of s
Figure Values of L for the M/M/s model for various values of s, the number of servers.

35. CFO Suggested Approach (Combine Into Teams)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).
The Chief Financial Officer has suggested combining the current one-person tech rep territories into larger territories that would be served jointly by multiple tech reps.
A territory with two tech reps:
Number of machines = 300 (versus 150 before)
Mean arrival rate = l = 6 (versus l = 3 before)
Mean service rate = m = 4 (as before)
Number of servers = s = 2 (versus s = 1 before)
Utilization factor = r = l/sm = 0.75 (as before)

36. M/M/s Model for the CFO’s Suggested Approach (Combine Into Teams of Two)
Figure This Excel template for the M/M/s model shows the results from applying this model to the approach suggested by Dupit’s chief financial officer with two tech reps assigned to each territory.

37. CFO Suggested Approach (Teams of Three)
The Chief Financial Officer has suggested combining the current one-person tech rep territories into larger territories that would be served jointly by multiple tech reps.
A territory with three tech reps:
Number of machines = 450 (versus 150 before)
Mean arrival rate = l = 9 (versus l = 3 before)
Mean service rate = m = 4 (as before)
Number of servers = s = 3 (versus s = 1 before)
Utilization factor = r = l/sm = 0.75 (as before)

38. M/M/s Model for the CFO’s Suggested Approach (Combine Into Teams of Three)
Figure This Excel template modifies the results in Figure 14.9 by assigning three tech reps to each territory.

39. Comparison of Wq with Territories of Different Sizes
Number of Tech Reps
Number of Machines l m s r Wq 1
150 3 4
0.75
0.75 workday (6 hours) 2
300 6
0.321 workday (2.57 hours)
450 9
0.189 workday (1.51 hours)
Table Comparison of Wq values with territories of different sizes for the Dupit problem.

40. Values of L for the M/D/s Model for Various Values of s
Figure Values of L for the M/D/s model for various values of s, the number of servers.

41. Values of L for the M/Ek/2 Model for Various Values of k
Figure Values of L for the M/Ek/2 model for various values of the shape parameter k.

42. The Four Approaches Under Considerations
Proposer
Proposal
Additional Cost
John Phixitt
Maintain one-person territories, but reduce number of machines assigned to each from 150 to 100
$300 million per year
VP for Engineering
Keep current one-person territories, but provide new state-of-the-art equipment to the tech-reps
One-time cost of $500 million
Chief Financial Officer
Change to three-person territories
None, except disadvantages of larger territories
Table The four approaches being considered by Dupit management.
Decision: Adopt the third proposal

43. Some Insights About Designing Queueing Systems
When designing a single-server queueing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system.
Decreasing the variability of service times (without any change in the mean) improves the performance of a queueing system substantially.
Multiple-server queueing systems can perform satisfactorily with somewhat higher utilization factors than can single-server queueing systems. For example, pooling servers by combining separate single-server queueing systems into one multiple-server queueing system greatly improves the measures of performance.
Applying priorities when selecting customers to begin service can greatly improve the measures of performance for high-priority customers.

44. Effect of High-Utilization Factors (Insight 1)
Figure This data table demonstrates Insight 1: When designing a single-server queueing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system.

45. Effect of Decreasing s (Insight 2)
Figure This data table demonstrates Insight 2: Decreasing the variability of service times (without any change in the mean) improves the performance of a queueing system substantially.

46. Economic Analysis of the Number of Servers to Provide
In many cases, the consequences of making customers wait can be expressed as a waiting cost.
The manager is interested in minimizing the total cost. TC = Expected total cost per unit time SC = Expected service cost per unit time WC = Expected waiting cost per unit time The objective is then to choose the number of servers so as to Minimize TC = SC + WC
When each server costs the same (Cs = cost of server per unit time), SC = Cs s
When the waiting cost is proportional to the amount of waiting (Cw = waiting cost per unit time for each customer), WC = Cw L

47. Acme Machine Shop
The Acme Machine Shop has a tool crib for storing tool required by shop mechanics.
Two clerks run the tool crib.
The estimates of the mean arrival rate l and the mean service rate (per server) m are l = 120 customers per hour m = 80 customers per hour
The total cost to the company of each tool crib clerk is $20/hour, so Cs = $20.
While mechanics are busy, their value to Acme is $48/hour, so Cw = $48.
Choose s so as to Minimize TC = $20s + $48L.

48. Excel Template for Choosing the Number of Servers
Figure This Excel template for using economic analysis to choose the number of servers with the M/M/s model is applied here to the Acme Machine Shop example with s = 3.

49. Comparing Expected Cost vs. Number of Clerks
Figure This data table compares the expected hourly costs with various alternative numbers of clerks assigned to the Acme Machine Shop tool crib.