1. Queuing Analysis
Chapter 13

2. Chapter Topics Elements of Waiting Line Analysis
The Single-Server Waiting Line System
Undefined and Constant Service Times
Finite Queue Length
Finite Calling Population
The Multiple-Server Waiting Line
Additional Types of Queuing Systems
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3. Overview
A significant amount of time is spent in waiting lines by people, products, etc.
Providing quick service is an important aspect of quality customer service.
The basis of waiting line analysis is the trade-off between the cost of improving service and the costs associated with making customers wait.
Queuing analysis is a probabilistic form of analysis.
The results are referred to as operating characteristics.
Results are used by managers of queuing operations to make decisions.
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4. Elements of Waiting Line Analysis (1 of 2)
Waiting lines form because people or things arrive at a service faster than they can be served.
Most operations have sufficient server capacity to handle customers in the long run.
Customers however, do not arrive at a constant rate nor are they served in an equal amount of time.
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5. Elements of Waiting Line Analysis (2 of 2)
Waiting lines are continually increasing and decreasing in length and approach an average rate of customer arrivals and an average service time in the long run.
Decisions concerning the management of waiting lines are based on these averages for customer arrivals and service times.
Averages are used in formulas to compute operating characteristics of the system which in turn form the basis of decision making.
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6. The Single-Server Waiting Line System (1 of 2)
Components of a waiting line system include arrivals (customers), servers, (cash register/operator), customers in line form a waiting line.
Factors to consider in analysis:
The queue discipline.
The nature of the calling population
The arrival rate
The service rate.
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7. The Single-Server Waiting Line System (2 of 2)
Figure 13.1 The Fast Shop Market waiting line system

8. Single-Server Waiting Line System Component Definitions
Queue Discipline: The order in which waiting customers are served.
Calling Population: The source of customers (infinite or finite).
Arrival Rate: The frequency at which customers arrive at a waiting line according to a probability distribution (frequently described by a Poisson distribution).
Service Rate: The average number of customers that can be served during a time period (often described by the negative exponential distribution).
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9. Single-Server Waiting Line System Single-Server Model
Assumptions of the basic single-server model:
An infinite calling population
A first-come, first-served queue discipline
Poisson arrival rate
Exponential service times
Symbols:
 = the arrival rate (average number of arrivals/time period)
 = the service rate (average number served/time period)
Customers must be served faster than they arrive ( < ) or an infinitely large queue will build up.
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10. Single-Server Waiting Line System
Basic Single-Server Queuing Formulas (1 of 2)
Probability that no customers are in the queuing system:
Probability that n customers are in the system:
Average number of customers in system:
Average number of customer in the waiting line:
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11. Single-Server Waiting Line System
Basic Single-Server Queuing Formulas (2 of 2)
Average time customer spends waiting and being served:
Average time customer spends waiting in the queue:
Probability that server is busy (utilization factor):
Probability that server is idle:
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12. Single-Server Waiting Line System
Operating Characteristics: Fast Shop Market (1 of 2)
 = 24 customers per hour arrive at checkout counter
 = 30 customers per hour can be checked out
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13. Single-Server Waiting Line System
Operating Characteristics for Fast Shop Market (2 of 2)
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14. Single-Server Waiting Line System
Steady-State Operating Characteristics
Because of the steady-state nature of operating characteristics:
Utilization factor, U, must be less than one:
U < 1, or  /  < 1 and  < .
The ratio of the arrival rate to the service rate must be less than one. In other words, the service rate must be greater than the arrival rate.
The server must be able to serve customers faster than the arrival rate in the long run, or waiting line will grow to infinite size.
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15. Single-Server Waiting Line System
Effect of Operating Characteristics (1 of 6)
A manager wishes to test several alternatives for reducing customer waiting time:
Addition of another employee to pack up purchases
Addition of another checkout counter.
Alternative 1: Addition of an employee
(raises service rate from  = 30 to  = 40 customers per hour).
Cost $150 per week, avoids loss of $75 per week for each minute of reduced customer waiting time.
System operating characteristics with new parameters:
Po = .40 probability of no customers in the system
L = 1.5 customers on average in the queuing system
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16. Single-Server Waiting Line System
Effect of Operating Characteristics (2 of 6)
System operating characteristics with new parameters (continued):
Lq = 0.90 customer on the average in the waiting line
W = hour average time in the system per customer
Wq = hour average time in the waiting line per customer
U = .60 probability that server is busy and customer must wait
I = .40 probability that server is available
Average customer waiting time reduced from 8 to 2.25 minutes worth $ per week.
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17. Single-Server Waiting Line System
Effect of Operating Characteristics (3 of 6)
Alternative 2: Addition of a new checkout counter ($6,000 plus $200 per week for additional cashier).
 = 24/2 = 12 customers per hour per checkout counter
 = 30 customers per hour at each counter
System operating characteristics with new parameters:
Po = .60 probability of no customers in the system
L = 0.67 customer in the queuing system
Lq = 0.27 customer in the waiting line
W = hour per customer in the system
Wq = hour per customer in the waiting line
U = .40 probability that a customer must wait
I = .60 probability that server is idle
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18. Single-Server Waiting Line System
Effect of Operating Characteristics (4 of 6)
Savings from the reduced waiting time worth:
$500 per week - $200 = $300 net savings per week.
After $6,000 is recovered, alternative 2 would provide:
$ = $18.75 more savings per week.
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19. Single-Server Waiting Line System
Effect of Operating Characteristics (5 of 6)
Table 13.1 Operating characteristics for each alternative system
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20. Single-Server Waiting Line System
Effect of Operating Characteristics (6 of 6)
Figure Cost trade-off for service levels
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21. Single-Server Waiting Line System
Solution with Excel and Excel QM (1 of 2)
Formula for Lq, average number in queue
=(1/(D4-D3))*60
=(D3/(D4*(D4-D3)))*60
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Exhibit 13.1

22. Single-Server Waiting Line System
Solution with Excel and Excel QM (2 of 2)
Click on “Add-Ins” to access the “Excel QM” menu
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Exhibit 13.2

23. Single-Server Waiting Line System Solution with QM for Windows
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Exhibit 13.3

24. Single-Server Waiting Line System Undefined and Constant Service Times
Constant, rather than exponentially distributed service times, occur with machinery and automated equipment.
Constant service times are a special case of the single-server model with undefined service times.
Queuing formulas for the undefined service time model:
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25. Single-Server Waiting Line System
Undefined Service Times Example (1 of 2)
Data: Single fax machine; arrival rate of 20 users per hour, Poisson distributed; undefined service time with mean of 2 minutes, standard deviation of 4 minutes.
Operating characteristics:
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26. Single-Server Waiting Line System
Undefined Service Times Example (2 of 2)
Operating characteristics (continued):
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27. Single-Server Waiting Line System Constant Service Times Formulas
In the constant service time model there is no variability in service times;  = 0.
Substituting  = 0 into equations:
All of the remaining formulas are the same as the single-server formulas.
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28. Single-Server Waiting Line System Constant Service Times Example
Car wash servicing one car at a time; constant service time of 4.5 minutes; arrival rate of customers of 10 per hour (Poisson distributed).
Determine average length of waiting line and average waiting time.
 = 10 cars per hour,  = 60/4.5 = 13.3 cars per hour
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29. Undefined and Constant Service Times Solution with Excel
Average number in the queue, Lq
=(D6/D3)*60
=D8+(1/D4)*60
Exhibit 13.4
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30. Undefined and Constant Service Times Solution with QM for Windows
Exhibit 13.5
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31. Finite Queue Length
In a finite queue, the length of the queue is limited.
Operating characteristics, where M is the maximum number in the system:
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32. Finite Queue Length Example (1 of 2)
Metro Quick Lube single bay service; space for one vehicle in service and three waiting for service; mean time between arrivals of customers is 3 minutes; mean service time is 2 minutes; both inter-arrival times and service times are exponentially distributed; maximum number of vehicles in the system equals 4.
Operating characteristics for  = 20,  = 30, M = 4:
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33. Finite Queue Length Example (2 of 2)
Average queue lengths and waiting times:
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34. Finite Queue Model Example Solution with Excel
Formula for P0 in cell D7
+((D3/D4)/(1-(D3/D4))) - ((D5+1)*(D3/D4) ^(D5+1))/(1-(D3/D4)^(D5+1))
Exhibit 13.6
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35. Finite Queue Model Example Solution with QM for Windows
Exhibit 13.7
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36. Finite Calling Population
In a finite calling population there is a limited number of potential customers that can call on the system.
Operating characteristics for a system with Poisson arrival and exponential service times:
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37. Finite Calling Population Example (1 of 2)
Wheelco Manufacturing Company; 20 machines; each machine operates an average of 200 hours before breaking down; average time to repair is 3.6 hours; breakdown rate is Poisson distributed, service time is exponentially distributed.
Is repair staff sufficient?
 = 1/200 hour = .005 per hour
 = 1/3.6 hour = per hour
N = 20 machines
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38. Finite Calling Population Example (2 of 2)
…The system seems woefully inadequate.
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39. Finite Calling Population Example
Solution with Excel and Excel QM (1 of 2)
Summation component for n=1 in cell G6
Array with summation components for P0 formula
P0 = 1/G26
Exhibit 13.8
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40. Finite Calling Population Example
Solution with Excel and Excel QM (2 of 2)
Click on “Add-Ins” to access the macro for the finite population model
Enter problem data in cells B7:B9
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Exhibit 13.9

41. Finite Calling Population Example Solution with QM for Windows
Exhibit 13.10
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42. Figure 13.3 A multiple-server waiting line
Multiple-Server Waiting Line (1 of 3)
Figure 13.3 A multiple-server waiting line
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43. Multiple-Server Waiting Line (2 of 3)
In multiple-server models, two or more independent servers in parallel serve a single waiting line.
Biggs Department Store service department; first-come, first-served basis.
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44. System at Biggs Department Store
Multiple-Server Waiting Line (3 of 3)
Customer Service
System at Biggs Department Store
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45. Multiple-Server Waiting Line Queuing Formulas (1 of 3)
Assumptions:
First-come first-served queue discipline
Poisson arrivals, exponential service times
Infinite calling population.
Parameter definitions:
 = arrival rate (average number of arrivals per time period)
 = the service rate (average number served per time period) per server (channel)
c = number of servers
c  = mean effective service rate for the system (must exceed arrival rate)
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46. Multiple-Server Waiting Line Queuing Formulas (2 of 3)
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47. Multiple-Server Waiting Line Queuing Formulas (3 of 3)
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48. Multiple-Server Waiting Line Biggs Department Store Example (1 of 2)
 = 10,  = 4, c = 3
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49. Multiple-Server Waiting Line Biggs Department Store Example (2 of 2)
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50. Multiple-Server Waiting Line Solution with Excel
Formula for P0
=((((D3)*(D4)*((D3/D4)^D5)*(D7))/(FACT
(D5-1)*(((D5*D4)-D3)^2))))+(D3/D4)
=(1/FACT(D5))*((D3/D4)^D5)*((D5*D4*D7)/((D5)*(D4)-(D3)))
Exhibit 13.11
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51. Multiple-Server Waiting Line Solution with Excel QM
Exhibit 13.12
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52. Multiple-Server Waiting Line Solution with QM for Windows
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Exhibit 13.13

53. Additional Types of Queuing Systems (1 of 2)
Figure 13.4 Single queues with single & multiple servers in sequence
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54. Additional Types of Queuing Systems (2 of 2)
Other items contributing to queuing systems:
Systems in which customers balk from entering system, or leave the line (renege).
Servers who provide service in other than a first-come, first-served manner
Service times that are not exponentially distributed or are undefined or constant
Arrival rates that are not Poisson distributed
Jockeying (i.e., moving between queues)
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55. Example Problem Solution (1 of 7)
Problem Statement: Citizens Northern Savings Bank loan officer customer interviews.
Customer arrival rate of four per hour, Poisson distributed; officer interview service time of 12 minutes per customer.
Determine operating characteristics for this system.
Add an additional officer creating a multiple-server queuing system with two channels. Determine operating characteristics for this system.
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56. Example Problem Solution (2 of 7)
Step 1: Determine Operating Characteristics for the Single-Server System
 = 4 customers per hour arrive,  = 5 customers per hour are served
Po = (1 -  / ) = ( 1 – 4 / 5) = .20 probability of no customers in the system
L =  / ( - ) = 4 / (5 - 4) = 4 customers on average in the queuing system
Lq = 2 / ( - ) = 42 / 5(5 - 4) = 3.2 customers on average in the waiting line
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57. Example Problem Solution (3 of 7)
Step 1 (continued):
W = 1 / ( - ) = 1 / (5 - 4) = 1 hour on average in the system
Wq =  / (u - ) = 4 / 5(5 - 4) = 0.80 hour (48 minutes) average time in the waiting line
Pw =  /  = 4 / 5 = .80 probability the new accounts officer is busy and a customer must wait
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58. Example Problem Solution (4 of 7)
Step 2: Determine the Operating Characteristics for the Multiple-Server System.
 = 4 customers per hour arrive;  = 5 customers per hour served; c = 2 servers
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59. Example Problem Solution (5 of 7)
Step 2: Determine the Operating Characteristics for the Multiple-Server System.
 = 4 customers per hour arrive;  = 5 customers per hour served; c = 2 servers
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60. Example Problem Solution (6 of 7)
Step 2 (continued):
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61. Example Problem Solution (7 of 7)
Step 2 (continued):
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