Queuing Models

1
Queuing Models
2
Introduction (1/2)

• Queuing is the study of waiting lines, or queues.
• The objective of queuing analysis is to design
  systems that enable organizations to perform
  optimally according to some criterion.
• Possible Criteria
• Maximum Profits.
• Desired Service Level.

3
Introduction (2/2)

• Analyzing queuing systems requires a clear
  understanding of the appropriate service
  measurement.
• Possible service measurements
• Average time a customer spends in line.
• Average length of the waiting line.
• The probability that an arriving customer must
  wait for service.

4
Elements of the Queuing Process

• A queuing system consists of three basic
  components
• Arrivals Customers arrive according to some
  arrival pattern.
• Waiting in a queue Arriving customers may have
  to wait in one or more queues for service.
• Service Customers receive service and leave the
  system.

5
The Arrival Process (1/2)

• There are two possible types of arrival processes
• Deterministic arrival process.
• Random arrival process.
• The random process is more common in businesses.

6
The Arrival Process (2/2)

• Under three conditions the arrivals can be
  modeled as a Poisson process
• Orderliness one customer, at most, will arrive
  during a predefined time interval.
• Stationarity for a given time frame, the
  probability of arrivals within a certain time
  interval is the same for all time intervals of
  equal length.
• Independence the arrival of one customer has no
  influence on the arrival of another.

7
The Poisson Arrival Process
(lt)ke- lt k!
P(X k)
Where l mean arrival rate per time
unit. t the length of the interval. e
2.7182818 (the base of the natural
logarithm). k! k (k -1) (k -2) (k -3) (3)
(2) (1).

8
Georges HARDWARE Arrival Process

• Customers arrive at Georges Hardware according
  to a Poisson distribution.
• Between 800 and 900 A.M. an average of 6
  customers arrive at the store.
• What is the probability that k customers will
  arrive between 800 and 830 in the morning (k
  0, 1, 2,)?

9
Georges HARDWARE An illustration of the
Poisson distribution.

• Input to the Poisson distribution
• l 6 customers per hour.t 0.5 hour.lt
  (6)(0.5) 3.

1
2
3
4
5
6
7
8
k
1
0
2
3
(lt) e- lt k !
1

0
P(X k )
2
0.049787
0.149361
0.224042
0.224042
3
0!
1!
2!
3!
10
The Waiting Line Characteristics

• Factors that influence the modeling of queues
• Line configuration
• Balking
• Reneging
• Jockeying

• Priority
• Tandem Queues
• Homogeneity

11
Line Configuration

• A single service queue.
• Multiple service queue with single waiting line.
• Multiple service queue with multiple waiting
  lines.
• Multistage service system.

12
Balking, Reneging, Jockeying

• Balking occurs if customers avoid joining the
  line when they perceive the line to be too long
• Reneging occurs when customers abandon the
  waiting line before getting served
• Jockeying (switching) occurs when customers
  switch lines once they perceived that another
  line is moving faster

13
Priority Rules

• These rules select the next customer for service.
• There are several commonly used rules
• First come first served (FCFS - FIFO).
• Last come first served (LCFS - LIFO).
• Estimated service time.
• Random selection of customers for service.

14
Multistage Service

• These are multi-server systems.
• A customer needs to visit several service
  stations (usually in a distinct order) to
  complete the service process.
• Examples
• Patients in an emergency room.
• Passengers prepare for the next flight.

15
Homogeneity

• A homogeneous customer population is one in which
  customers require essentially the same type of
  service.
• A non-homogeneous customer population is one in
  which customers can be categorized according to
• Different arrival patterns
• Different service treatments.

16
The Service Process

• In most business situations, service time varies
  widely among customers.
• When service time varies, it is treated as a
  random variable.
• The exponential probability distribution is used
  sometimes to model customer service time.

17
The Exponential Service Time Distribution
m the average number of customers who
can be served per time period. Therefore, 1/m
the mean service time.
18
Schematic illustration of the exponential
distribution
The probability that service is completed within
t time units P(X t) 1 - e-mt
X t
19
Georges HARDWARE Service time

• George estimates the average service time to be
  1/m 4 minutes per customer.
• Service time follows an exponential distribution.
• What is the probability that it will take less
  than 3 minutes to serve the next customer?

20
GEORGEs HARDWARE

• The mean number of customers served per minute is
  ¼ ¼(60) 15 customers per hour.
• P(X lt .05 hours) 1 e-(15)(.05) 0.5276

21
GEORGEs HARDWARE Using Excel for the
Exponential Probabilities
EXPONDIST(A10,B3,FALSE) Drag to B11B26
22
The Exponential Distribution - Characteristics

• The memoryless property (Markov)
• No additional information about the time left for
  the completion of a service, is gained by
  recording the time elapsed since the service
  started.
• For Georges, the probability of needing more
  than 3 minutes is (1-0.527630.47237) independent
  of how long the customer has been served already.
• P(Tst / Ts) P(Tt)
• The Exponential and the Poisson distributions are
  related to one another.
• If customer arrivals follow a Poisson
  distribution with mean rate l, their interarrival
  times are exponentially distributed with mean
  time 1/l.

23
Performance Measures (1/4)

• Performance can be measured by focusing on
• Customers in queue.
• Customers in the system.
• Performance is measured for a system in steady
  state.

24
Performance Measures (2/4)

• The transient period occurs at the initial time
  of operation.
• Initial transient behavior is not indicative of
  long run performance.

n
Roughly, this is a transient period
Time
25
Performance Measures (3/4)

• The steady state period follows the transient
  period.

n
This is a steady state period..
Roughly, this is a transient period

• Meaningful long run performance measures can be
  calculated for the system when in steady state.

Time
26
Performance Measures (4/4)

• In order to achieve steady state, theeffective
  arrival rate must be less than the sum of the
  effective service rates .

llt km Each with service rate of m
llt m1 m2mk For k servers with service rates mi
llt m For one server
27
Steady State Performance Measures
P0 Probability that there are no customers
in the system.
Pn Probability that there are n customers
in the system.
L Average number of customers in the
system.
Lq Average number of customers in the queue.
W Average time a customer spends in the
system.
Wq Average time a customer spends in the queue.
Pw Probability that an arriving customer
must wait for service.
r Utilization rate for each server (the
percentage of time that each server is busy).
28
Littles Formulas

• Littles Formulas represent important
  relationships between L, Lq, W, and Wq.
• These formulas apply to systems that meet the
  following conditions
• Single queue systems,
• Customers arrive at a finite arrival rate l,
  and
• The system operates under a steady state
  condition.
• L l W Lq l Wq L Lq l/m

For the case of an infinite population
29
Classification of Queues

• Queuing system can be classified by
• Arrival process.
• Service process.
• Number of servers.
• System size (infinite/finite waiting line).
• Population size.
• Notation
• M (Markovian) Poisson arrivals or exponential
  service time.
• D (Deterministic) Constant arrival rate or
  service time.
• G (General) General probability for arrivals or
  service time.

Example M / M / 6 / 10 / 20
30
M/M/1 Queuing System - Assumptions

• Poisson arrival process.
• Exponential service time distribution.
• A single server.
• Potentially infinite queue.
• An infinite population.

31
M / M /1 Queue - Performance Measures

• P0 1 (l/m)
• Pn 1 (l/m)(l/m)n
• L l /(m l)
• Lq l2 /m(m l)
• W 1 /(m l)
• Wq l /m(m l)
• Pw l / m
• r l / m

The probability that a customer waits in the
system more than t is P(Xgtt) e-(m - l)t
32
MARYs SHOES

• Customers arrive at Marys Shoes every 12 minutes
  on the average, according to a Poisson process.
• Service time is exponentially distributed with an
  average of 8 minutes per customer.
• Management is interested in determining the
  performance measures for this service system.

33
MARYs SHOES - Solution

• Input
• l 1/12 customers per minute 60/12 5 per
  hour.
• m 1/ 8 customers per minute 60/ 8 7.5
  per hour.
• Performance Calculations

P0 1 - (l/m) 1 - (5/7.5) 0.3333 Pn 1 -
(l/m)(l/m)n (0.3333)(0.6667)n
L l/(m - l)
2 Lq l2/m(m - l) 1.3333 W 1/(m - l)
0.4 hours 24 minutes Wq l/m(m - l)
0.26667 hours 16 minutes
34
M/M/k Queuing Systems

• Characteristics
• Customers arrive according to a Poisson process
  at a mean rate l.
• Service times follow an exponential distribution.
• There are k servers, each of who works at a rate
  of m customers (with kmgt l).
• Infinite population, and possibly infinite line.

35
M / M /k Queue - Performance Measures
36
M / M /k Queue - Performance Measures
The performance measurements L, Lq, Wq,, can be
obtained from Littles formulas.
37
LITTLE TOWN POST OFFICE

• Little Town post office is open on Saturdays
  between 900 a.m. and 100 p.m.
• Data
• On the average 100 customers per hour visit the
  office during that period. Three clerks are on
  duty.
• Each service takes 1.5 minutes on the average.
• Poisson and Exponential distributions describe
  the arrival and the service processes
  respectively.

38
LITTLE TOWN POST OFFICE

• The Postmaster needs to know the relevant service
  measures in order to
• Evaluate the current service level.
• Study the effects of reducing the staff by one
  clerk.

39
LITTLE TOWN POST OFFICE - Solution

• This is an M / M / 3 queuing system.
• Input
• l 100 customers per hour.
• m 40 customers per hour (60/1.5).
• Does steady state exist (l lt km )?
• l 100 lt km 3(40) 120.

40
LITTLE TOWN POST OFFICE solution continued

• First P0 is found by

• P0 is used to determine all the other performance
  measures.

41
LITTLE TOWN POST OFFICE
42
LITTLE TOWN POST OFFICE
43
M/G/1 Queuing System

• Assumptions
• Customers arrive according to a Poisson process
  with a mean rate l.
• Service time has a general distribution with mean
  rate m.
• One server.
• Infinite population, and possibly infinite line.

44
M/G/1 Queuing System Pollaczek - Khintchine
Formula for L
Note It is not necessary to know the particular
service time distribution. Only the mean and
standard deviation of the distribution are needed.
45
VASSILISS TV REPAIR SHOP (1/2)

• Vassilis repairs television sets and VCRs.
• Data
• It takes an average of 2.25 hours to repair a
  set.
• Standard deviation of the repair time is 45
  minutes.
• Customers arrive at the shop once every 2.5 hours
  on the average, according to a Poisson process.
• Vassilis works 9 hours a day, and has no help.
• He considers purchasing a new piece of equipment.
• New average repair time is expected to be 2
  hours.
• New standard deviation is expected to be 40
  minutes.

46
VASSILISS TV REPAIR SHOP (2/2)
Vassilis wants to know the effects of using the
new equipment on 1. The average number of
sets waiting for repair 2. The average time a
customer has to wait to get his repaired
set.
47
VASSILISS TV REPAIR SHOP - Solution

• This is an M/G/1 system (service time is not
  exponential (note that s ¹ 1/m).
• Input
• The current system (without the new equipment)
• l 1/ 2.5 0.4 customers per hour.
• m 1/ 2.25 0.4444 costumers per hour.
• s 45/ 60 0.75 hours.
• The new system (with the new equipment)
• m 1/2 0.5 customers per hour.
• s 40/ 60 0.6667 hours.

48
VASSILISS TV REPAIR SHOP - Results
49
M / M / k / F Queuing System

• Many times queuing systems have designs that
  limit their line size.
• When the potential queue is large, an infinite
  queue model gives accurate results, even though
  the queue might be limited.
• When the potential queue is small, the limited
  line must be accounted for in the model.

50
Characteristics of M/M/k/F Queuing System

• Poisson arrival process at mean rate l.
• k servers, each having an exponential service
  time with mean rate m.
• Maximum number of customers that can be present
  in the system at any one time is F.
• Customers are blocked (and never return) if the
  system is full.

51
M/M/k/F Queuing System Effective Arrival Rate

• A customer is blocked if the system is full.
• The probability that the system is full is PF
  (100PF of the arriving customers do not enter
  the system).
• The effective arrival rate the rate of arrivals
  that make it through into the system (le).

52
ANTONISS ROOFING COMPANY (1/3)

• Antonis gets most of its business from customers
  who call and order service.
• When a telephone line is available but the
  secretary is busy serving a customer, a new
  calling customer is willing to wait until the
  secretary becomes available.
• When all the lines are busy, a new calling
  customer gets a busy signal and calls a
  competitor.

53
Antoniss ROOFING COMPANY (2/3)

• Data
• Arrival process is Poisson, and service process
  is Exponential.
• Each phone call takes 3 minutes on the average.
• 10 customers per hour call the company on the
  average.
• One appointment secretary takes phone calls from
  3 telephone lines.

54
ANTONISS ROOFING COMPANY (3/3)

• Management would like to design the following
  system
• The fewest lines necessary.
• At most 2 of all callers get a busy signal.
• Management is interested in the following
  information
• The percentage of time the secretary is busy.
• The average number of customers kept on hold.
• The average time a customer is kept on hold.
• The actual percentage of callers who encounter
  a busy signal.

55
ANTONISS ROOFING COMPANY - Solution

• This is an M/M/1/3 system
• Input
• l 10 per hour.
• m 20 per hour (1/3 per minute).
• Excel spreadsheet gives
• P0 0.533, P1 0.133, P3 0.06
• 6.7 of the customers get a busy signal.
• This is above the goal of 2.

M/M/1/4 system
M/M/1/5 system
See spreadsheet next
P0 0.516, P1 0.258, P2 0.129, P3 0.065,
P4 0.032 3.2 of the customers get the
busy signal Still above the goal of 2
56
ANTONISS ROOFING COMPANY - Spreadsheet
Solution
57
Probability Analysis, F1, 2, 3, 4, 5
58
Sensitivity Analysis for F
59
M / M / 1 / / m Queuing Systems

• In this system the number of potential customers
  is finite and relatively small.
• As a result, the number of customers already in
  the system affects the rate of arrivals of the
  remaining customers.
• Characteristics
• A single server.
• Exponential service time, Poisson arrival
  process.
• A population size of a (finite) m customers.

60
Progress Homes

• Progress Homes runs four different development
  projects.
• Data
• At each site running a project is interrupted
  once every 20 working days on the average.
• The V.P. for construction handles each stoppage.
• How long on the average a site is
  non-operational?
• If it takes 2 days on the average to restart a
  projects progress (the V.P. is using the current
  car).
• If it takes 1.875 days on the average to restart
  a projects progress (the V.P. is using a new car)

61
Progress Homes Solution

• This is an M/M/1//4 system, where
• The four sites are the four customers.
• The V.P. for construction is the server.
• Input
• l 0.05 (1/20)
• m 0.5 m 0.533

(1/2 days, using the current car)
(1/1.875 days, using a new car).
62
Progress Homes Computer Output
63
Progress Homes Summary of Results
64
Economic Analysis of Queuing Systems

• The performance measures previously developed are
  used next to determine a minimal cost queuing
  system.
• The procedure requires estimated costs such as
• Hourly cost per server .
• Customer goodwill cost while waiting in line.
• Customer goodwill cost while being served.

65
WILSON FOODS (1/2)

• Wilson Foods has an 800 number to answer
  customers questions.
• If all the customer representatives are busy when
  a new customer calls, he/she is asked to stay on
  the line.
• A customer stays on the line if the waiting time
  is not longer than 3 minutes.

66
WILSON FOODS (2/2)

• Data
• On the average 225 calls per hour are received.
• An average phone call takes 1.5 minutes.
• A customer will stay on the line waiting at most
  3 minutes.
• A customer service representative is paid 16 per
  hour.
• Wilson pays the telephone company 0.18 per
  minute when the customer is on hold or when being
  served.
• Customer goodwill cost is 0.20 per minute while
  on hold.
• Customer goodwill cost while in service is 0.05.

How many customer service representatives
should be used to minimize the hourly cost of
operation?
67
WILSON FOODS Solution

• The total hourly cost model

Average hourly goodwill cost for customers on
hold
Total hourly wages
Total average hourly Telephone charge
Average hourly goodwill cost for customers in
service
TC(K) Cwk (Ct gs)Lq (Ct gs)(L Lq)
68
WILSON FOODS Solution continued

• Input
• Cw 16
• Ct 10.80 per hour 0.18(60)
• gw 12 per hour 0.20(60)
• gs 3 per hour 0.05(60)
• The Total Average Hourly Cost
• TC(K) 16K (10.83)L (12 - 3)Lq 16K
  13.8L 9Lq

69
WILSON FOODS Solution re-continued

• Assuming a Poisson arrival process and an
  Exponential service time, we have an M/M/K
  system.
• l 225 calls per hour.
• m 40 per hour (60/1.5).
• The minimal possible value for K is 6 to ensure
  that steady state exists (lltKm).

70
WILSON FOODS Solution re-re-continued

• Summary of results of the runs for k6,7,8,9,10

Conclusion employ 8 customer service
representatives.
71
WILSON FOODS Results for k6
72
WILSON FOODS Sensitivity Analysis