1. Service Operations Management (SOM) Waiting Line Models
Services and Responsiveness
Waiting Line Models
Example
Compiled by:
Alex J. Ruiz-Torres, Ph.D.
From information developed by many.

2. Services and Responsiveness
Responsiveness is a dimension of any service system/product.
Responsive: Speed of service/ time to complete the process.
Two components
Time waiting for the service (to start).
Time to perform the service.
For example.
Time waiting to get an oil change.
Time it takes to perform the oil change.

3. Services and Responsiveness
Time waiting for a service depends on two factors:
Volume of customers.
Number of resources available.
Given a fixed number of resources, as the number of customer arriving increases, so does the average time waiting.
Given a fixed number of customers arriving, as the number of resources available increases, the average waiting time of the customers decreases.

4. Services and Responsiveness
Waiting Lines are present everywhere (banks, supermarkets, airports, …)
Can be in the form of people, documents, phone calls, vehicles, packages, electronic requests, …waiting.
These are types of “customers”
planes waiting to takeoff
the line at your local CESCO

5. Services and Responsiveness
In general, no customer likes to wait.
Customers that wait may not return (loss of future sales).
Customers with significant dissatisfaction due to long waits will tell others (loss of future sales from other customers).
a waiting line does not have to be an actual line (turns)

6. Services and Responsiveness
As a system, waiting time = non-productive time = cost.
A company truck waiting to be maintained can represent several thousand dollars on lost revenue (if on the road performing work for example).
Ships waiting in line to enter the Panama Canal cost thousands of dollars in lost productive time.
Point: waiting time = cost

7. Services and Responsiveness
Responsiveness is related to the resources available (the service capacity).
One teller in the bank = long waits.
Four tellers in the bank = never a wait.
But resource costs are about four times larger in the second case.
Resource costs relate to the variable and fixed cost associated with the facilities, technology and people used to offer the service.
Management needs to determine the “right”service capacity to balance resource and waiting costs.

8. Services and Responsiveness
Total costs
Resource costs $
Waiting costs
Service Capacity
(Resources available)
As the service capacity increases the resource cost increase.
As the service capacity increases, the waiting time decreases. Therefore the waiting cost decrease.
The goal is to find the right amount of service capacity where the total costs are minimized.

9. Waiting Line Models
Models that predict the behavior of systems where lines form. Also called queueing theory models. (queue = line).
Basic assumptions: random arrivals of customers, limited capacity, and random process times.
Conceptual/ based on simple equations.
Many simplifying assumptions, for example no customers ever leave the line.
But its helpful as it provide some guidelines; other more sophisticated models can be used.

10. Waiting Line Models The components:
Entities: what is being processed (people, cars, packages, phone calls, …)
Demand rate : entities arrive over time to be served (order a product, be inspected, …)
Server: the resource (s) that completes the task (people, machines)
Service time: the time to complete the task
Waiting line: where entities wait for service

11. Waiting Line Models Three critical parameters
Number of servers (resources). This is identified as S.
The arrival rate: how many customers arrive per period of time. This is identified as l (lambda)
The service rate: how many customers can be served per period of time. This is identified as m (mu)

12. Waiting Line Models Examples for l Examples for m
11 customer arrive each hour for an oil change.
2,500 flights arrive per day at the Atlanta airport
Examples for m
Each oil changing station can perform 4 oil changes per hour.
The Atlanta airport can serve 3,000 flights per day

13. Waiting Line Models
Arrival rate is the inverse of the inter-arrival time
For example a clinic receives a patient on average every 5 minutes.
Interarrival time = 5 𝑚𝑖𝑛𝑢𝑡𝑒 𝑝𝑎𝑡𝑖𝑒𝑛𝑡
Inverse: Arrival rate l= 1 𝑝𝑎𝑡𝑖𝑒𝑛𝑡 5 𝑚𝑖𝑛𝑢𝑡𝑒 = 0.2 𝑝𝑎𝑡𝑖𝑒𝑛𝑡 𝑚𝑖𝑛𝑢𝑡𝑒
We can convert to hours to make it more intuitive
= 1 𝑝𝑎𝑡𝑖𝑒𝑛𝑡 5 𝑚𝑖𝑛𝑢𝑡𝑒 × 60 𝑚𝑖𝑛𝑢𝑡𝑒 1 ℎ𝑜𝑢𝑟 = 𝟏𝟐 𝒑𝒂𝒕𝒊𝒆𝒏𝒕 𝒉𝒐𝒖𝒓

14. Waiting Line Models Service rate is the inverse of the service time.
For example, it takes on average 25 minutes to perform an oil change on a car.
Service time = 25 𝑚𝑖𝑛𝑢𝑡𝑒 𝑐𝑎𝑟
Inverse: service rate m= 1 𝑐𝑎𝑟 25 𝑚𝑖𝑛𝑢𝑡𝑒 = 𝑐𝑎𝑟 𝑚𝑖𝑛𝑢𝑡𝑒
We can convert to hours to make it more intuitive
= 1 𝑐𝑎𝑟 25 𝑚𝑖𝑛𝑢𝑡𝑒 × 60 𝑚𝑖𝑛𝑢𝑡𝑒 1 ℎ𝑜𝑢𝑟 = 𝟐.𝟒 𝒄𝒂𝒓 𝒉𝒐𝒖𝒓

15. Waiting Line Models Model provides characteristics of the line
For S = 1 (single server)
Probability that there are y entities in the line.
Probability of 0 entities in the line
P0 = 1 – λ/μ
Probability of n entities in the line
Pn = (λ/μ)n P0

16. Example A small hair saloon receives about 7 clients per 8 hour day.
A stylist takes on average 46 minutes per client.
Questions of interest.
What is the probability there is nobody in line?
What is the probability at most two customers waiting?
What about the probability there are 4 or more customers waiting?

17. Example l = 7 𝑐𝑙𝑖𝑒𝑛𝑡 𝑑𝑎𝑦 𝑑𝑎𝑦 8 ℎ𝑜𝑢𝑟 = 0.875 𝑐𝑙𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟
First determine l and m. Useful in this case to convert to hours.
l = 7 𝑐𝑙𝑖𝑒𝑛𝑡 𝑑𝑎𝑦 𝑑𝑎𝑦 8 ℎ𝑜𝑢𝑟 = 𝑐𝑙𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟
to determine m, need to calculate the inverse of the service time.
the service time = 46 𝑚𝑖𝑛𝑢𝑡𝑒 𝑐𝑙𝑖𝑒𝑛𝑡
m = 𝑐𝑙𝑖𝑒𝑛𝑡 46 𝑚𝑖𝑛𝑢𝑡𝑒 ( 60 𝑚𝑖𝑛𝑢𝑡𝑒 ℎ𝑜𝑢𝑟 )= 𝑐𝑙𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟

18. Example What is the probability there is nobody in line?
That is P0 = 1 – λ/μ
= 1 – = = 32.9%
What about the probability at most two customers waiting?
= P0 + P1 + P2
Need to use this equation: Pn = (λ/μ)n P0 =
= 69.8%
P1 = 0.221
P2 = 0.148

19. Example What about the P there are 4 or more customers waiting?
= P4 + P5 + P6 + … P
= 1 – (P0 + P1 + P2 + P3)
Need to use this equation: Pn = (λ/μ)n P0
= 1 – ( )
= 1 – = = 20.3%
P1 = 0.221
P2 = 0.148
P3 = 0.099

20. Example A small hair saloon receives about 7 clients per 8 hour day.
A stylist takes on average 46 minutes per client.
Questions of interest.
How many customers are waiting on average?
What is the average waiting time per customer?

21. Waiting Line Models
Model provides steady state characteristics of the waiting line
Average line size (number in line): Lq
Average waiting time: Wq = Lq / l

22. Waiting Line Models Lq = Average line size A view of what is Lq
= Avg(3, 6, 5, 3, 1, 0, 3) = 3
time

23. Waiting Line Models Lq = Average line size A view of what is Lq
= Avg(1, 3, 0, 0, 0, 0, 1) = 0.57
time

24. Waiting Line Models Wq = Average wait time A view at Wq
customer 001. Arrived and nobody in line. Wait time = 0
customer 002. Arrived and had to wait. Wait time = 6
customer 003. Arrived and had to wait. Wait time = 8
customer 004. Arrived and had to wait. Wait time = 2
customer 005. Arrived and nobody in line. Wait time = 0
customer 006. Arrived and had to wait. Wait time = 1
Wq = Average wait time
= Avg(0, 6, 8, 2, 0, 1) = 2.83

25. Waiting Line Models There are equations to find Lq.
However, we will use the Lq Table (precalculated values)
Table provides the average length of the line given the l/m ratio and the number of servers S.

26. Example
What is the performance of the system with a single stylist in terms of the size of the line and the waiting time?
l / m = 𝟎.𝟖𝟕𝟓 𝟏.𝟑𝟎𝟒 = 0.671, S = 1
Lq = 1.63 (average line is between 1 and 2 customers)
Wq = Lq / l = 𝟏.𝟔𝟑 𝒄𝒍𝒊𝒆𝒏𝒕 𝟎.𝟖𝟕𝟓 𝒄𝒍𝒊𝒆𝒏𝒕/𝒉𝒐𝒖𝒓
Wq = hours
The average number of customers in line is 1.63 and the average time each customers waits is (1:52) hours.

27. Example Second stylist? Lq = 0.1
l / m = 𝟎.𝟖𝟕𝟓 𝟏.𝟑𝟎𝟒 = 0.671, S = 2
Lq = 0.1
Wq = Lq / l = 𝟎.𝟏 𝒄𝒍𝒊𝒆𝒏𝒕 𝟏.𝟑𝟎𝟒 𝒄𝒍𝒊𝒆𝒏𝒕/𝒉𝒐𝒖𝒓
Wq = hours = 4.6 minutes
The average number of customers in line is 0.1 (there is almost never a line) and the average time each customers waits is 4.6 minutes.

28. Waiting Line Models
Model provides a way to consider the tradeoffs between waiting and service costs.
Key variable.
wcr = waiting cost rate. The cost per time unit waiting (lost sales, lost revenue, lost reputation,…)
WCe = waiting cost per entity
= wcr  Wq
RCe = Resource cost per entity
TCe = Total costs per entity
= WCe + RCe

29. Example
Costs
Every hour a customer waits represents $15 in lost future sales. wcr = $15/hour.
Fixed costs = $12/hour
Labor and other variable costs associated with each stylist = $28/hour.
Case Wq
WCe
$15 x Wq
RC_hour
RCe
RC_hour / l
TCe
S = 1
1.863 hrs.
$27.94
$12 + $28 = $40
$45.71
$73.65
S = 2
hrs.
$1.15
$12 + $56 = $68
$77.71
$78.86

30. Example
What if waiting costs are being underestimated.
wcr = $50/hour.
Case Wq
WCe
$50 x Wq
RC_hour
RCe
RC_hour / l
TCe
S = 1
1.863 hrs.
$93.15
$12 + $28 = $40
$45.71
$138.86
S = 2
hrs.
$3.84
$12 + $56 = $68
$77.71
$81.55

31. Queue Management QM Systems
Software / technology that controls queues based on real time information.