1 Queuing Systems (2). Queueing Models (Henry C. Co)2 Queuing Analysis Cost of service capacity Cost...
-
Upload
miles-richardson -
Category
Documents
-
view
217 -
download
0
Transcript of 1 Queuing Systems (2). Queueing Models (Henry C. Co)2 Queuing Analysis Cost of service capacity Cost...
1
Queuing Systems (2)
Queueing Models (Henry C. Co) 2
Queuing Analysis
Cost ofservicecapacity
Cost ofcustomerswaiting
Co
st
Service capacity
Totalcost
Customerwaiting cost
Capacitycost= +
Queueing Models (Henry C. Co) 3
The Basic Model
Queueing Models (Henry C. Co) 4
A Basic Queue
Server
Stephen R. LawrenceStephen R. LawrenceLeeds School of BusinessUniversity of ColoradoBoulder, CO 80309-0419
Queueing Models (Henry C. Co) 5
A Basic Queue
CustomerArrivals Server
Queueing Models (Henry C. Co) 6
A Basic Queue
Server
Queueing Models (Henry C. Co) 7
A Basic Queue
CustomerDepartures
Server
Queueing Models (Henry C. Co) 8
A Basic Queue
Queue(waiting line)Customer
Arrivals
CustomerDepartures
Server
Queueing Models (Henry C. Co) 9
A Basic Queue
Queue(waiting line)Customer
Arrivals
CustomerDepartures
Server
Line too long?Customer balks
(never enters queue)
Queueing Models (Henry C. Co) 10
A Basic Queue
Queue(waiting line)Customer
Arrivals
CustomerDepartures
Line too long?Customer reneges(abandons queue)
Server
Line too long?Customer balks
(never enters queue)
Queueing Models (Henry C. Co) 11
Queuing Analysis
Single Channel (or Single Server) Queue
Queueing Models (Henry C. Co) 12
Queuing Analysis
Service
Rate (
Queueing Models (Henry C. Co) 13
Queuing Analysis
ArrivalRate (
Service
Rate (
Queueing Models (Henry C. Co) 14
Queuing Analysis
ArrivalRate (
Average Waiting
Time in Queue (Wq )Service
Rate (
Queueing Models (Henry C. Co) 15
Queuing Analysis
Arrival
Rate (Average Number of
People in Queue (Lq )
Average Waiting
Time in Queue (Wq )Service
Rate (
Queueing Models (Henry C. Co) 16
Queuing Analysis
Arrival
Rate (Average Number of
People in Queue (Lq )
Average Time in System (W )
Average Number in System (L )
Average Waiting
Time in Queue (Wq )Service
Rate (
Queueing Models (Henry C. Co) 17
Characteristics of a Queue
Queueing Models (Henry C. Co) 18
Source population Arrival characteristics Physical features of lines Selection from the waiting line Service facility Exit
Elements of Queuing System
Arrivals ServiceWaitingline
Exit
Processingorder
System
Queueing Models (Henry C. Co) 19
Source Population May be finite or infinite. For practical intent and purposes,
when the population is large in comparison to the service system, we assume the source population to be infinite (e.g., in a small barber shop, 200 potential customers per day may be treated as an infinite population).
Queueing Models (Henry C. Co) 20
Arrival Pattern of arrivals
Controllable arrival pattern Movie theatres offering Monday specials. Department stores running sales. Airlines offering off-season rates. Overseas telecom rates from 1:00 a.m. To 7:00
a.m. Uncontrollable arrival pattern
Emergency operations. Fire department.
Size of arrivals: single or batch arrival? Probability distribution pattern of arrivals.
Periodic: constant time-between-arrivals (TBA). Purely random TBA (e.g., exponential distribution).
Queueing Models (Henry C. Co) 21
Degree of patients A patient arrival is one who waits as long
as necessary until the service facility is ready to serve him/her (even if the customer grumble and behave discourteously or Impatiently).
Impatient arrivals. Balking: views the situation (length of queue)
and then decides to leave. Reneging: views the situation, joins the
queue, after some time, departs without being served.
Queueing Models (Henry C. Co) 22
Physical Features of Waiting Line Length of line: infinite or finite waiting
capacity? Number of lines; configuration of the
lines; jockeying.
Queueing Models (Henry C. Co) 23
Selection from the Waiting Line Queue discipline: priority rule(s) for
determining the order of service to customers in a waiting line FIFO. By reservations/appointment only/first. Emergencies first. Highest profit customer first. Largest orders first. Best customer first. Longest waiting time in line first. Soonest promised date first. Shortest processing time first.
Line structuring: express checkouts (supermarkets); “commercial transactions only” (banks).
Queueing Models (Henry C. Co) 24
Service Facility Structure
Single-channel single-phase.
Single-channel multi-phase.
Multi-channel single-phase.
Multi-channel multi-phase.
Mixed.
Service rate Constant Random (probability
distribution).
Queuing SystemsMultiple channel
Multiple phase
Queueing Models (Henry C. Co) 25
Exit Return to source population
Recurring-common-cold case. Low probability of re-service
Appendectomy-only-once case.
Queueing Models (Henry C. Co) 26
Steady State
Queueing Models (Henry C. Co) 27
A stable system: The queue will never increase to infinity. An empty state is reached for sure after some time period.
Condition for Stability: >. This condition MUST be met to make all formulas valid.
The steady state: Probability {n customers in the system} does not depend on the time.
Queueing Models (Henry C. Co) 28
Waiting Time vs Utilization
System Utilization
Ave
rag
e n
um
ber
on
tim
e w
aiti
ng
in
lin
e
0 100%
Queueing Models (Henry C. Co) 29
M/M/1 Queues
1st M (for “Markovian) – Arrival Distribution is Exponential2nd M – Service Distribution is Exponential1 – Single Channel
Queueing Models (Henry C. Co) 30
Population Time horizon: an infinite horizon. Source Population: infinite.
Queueing Models (Henry C. Co) 31
Arrival Process The inter-arrival time is an
exponentially-distributed random variable with average arrival rate = .
If the inter-arrival time is an exponentially-distributed random variable, then the number of arrivals during the fixed period of time is a Poisson distribution.
No balking or reneging
Queueing Models (Henry C. Co) 32
00.05
0.10.15
0.20.25
0 1 2 3 4 5 6 7 8 9 10 11 12
Poisson Distribution
Queueing Models (Henry C. Co) 33
Service Process The service time is also assumed to be
exponentially distributed with mean service rate .
Only 1 server First-come-first-served (FCFS) queue
priority Mean length of service = 1/ No limit on the queue size.
Queueing Models (Henry C. Co) 34
Operating Characteristics
Utilization (fraction of time server is busy)
Queueing Models (Henry C. Co) 35
Operating Characteristics
Utilization (fraction of time server is busy)
Expected (Average) waiting times
W 1
W Wq
Queueing Models (Henry C. Co) 36
Operating Characteristics
Utilization (fraction of time server is busy)
Average waiting times
Average numbers
W 1
W Wq
L LLq
Queueing Models (Henry C. Co) 37
Fundamental Relationship
Little’s Law: L=W or Lq= Wq
Queueing Models (Henry C. Co) 38
Example
Stephen R. LawrenceLeed School of BusinessUniversity of ColoradoBoulder, CO 80309-0419
Queueing Models (Henry C. Co) 39
Example
Boulder Reservoir has one launching ramp for small boats.On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.
Queueing Models (Henry C. Co) 40
Example
Boulder Reservoir has one launching ramp for small boats.On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.
= 6/hr = 1/s =1/6 = 0.167/min or 10/hr
Queueing Models (Henry C. Co) 41
Example
Boulder Reservoir has one launching ramp for small boats.On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.
= 6/hr = 1/s =1/6 = 0.167/min or 10/hr
= 6/10 = 0.6 or 60%
Queueing Models (Henry C. Co) 42
Example
Boulder Reservoir has one launching ramp for small boats.On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.
= 6/hr = 1/s =1/6 = 0.167/min or 10/hr
= 6/10 = 0.6 or 60%
L = = 6/(10-6) = 1.5 boatsLq = L = 1.5(0.6) = 0.9 boats
Queueing Models (Henry C. Co) 43
Example
Boulder Reservoir has one launching ramp for small boats.On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.
= 6/hr = 1/s =1/6 = 0.167/min or 10/hr
= 6/10 = 0.6 or 60%
L = = 6/(10-6) = 1.5 boatsLq = L = 1.5(0.6) = 0.9 boats
W = 1/= 1/(10-6) = 0.25 hrs or 15 minsWq = W = 0.25(0.6) = 0.15 hrs or 9 mins
Queueing Models (Henry C. Co) 44
Example (cont.)
During the busy Fourth of July weekend, boats are expectedto arrive at an average rate of 9 per hour.
Queueing Models (Henry C. Co) 45
Example (cont.)
During the busy Fourth of July weekend, boats are expectedto arrive at an average rate of 9 per hour.
= 9/hr = 1/s =1/6 = 0.167/min or 10/hr
Queueing Models (Henry C. Co) 46
Example (cont.)
During the busy Fourth of July weekend, boats are expectedto arrive at an average rate of 9 per hour.
= 9/hr = 1/s =1/6 = 0.167/min or 10/hr
= 9/10 = 0.9 or 90%
Queueing Models (Henry C. Co) 47
Example (cont.)
During the busy Fourth of July weekend, boats are expectedto arrive at an average rate of 9 per hour.
= 9/hr = 1/s =1/6 = 0.167/min or 10/hr
= 9/10 = 0.9 or 90%
L = = 9/(10-9) = 9.0 boatsLq = L = 9(0.6) = 5.4 boats
Queueing Models (Henry C. Co) 48
Example (cont.)
During the busy Fourth of July weekend, boats are expectedto arrive at an average rate of 9 per hour.
= 9/hr = 1/s =1/6 = 0.167/min or 10/hr
= 9/10 = 0.9 or 90%
L = = 9/(10-9) = 9 boatsLq = L = 9(0.6) = 5.4 boats
W = 1/= 1/(10-9) = 1.0 hrs or 60 minsWq = W = 1(0.9) = 0.9 hrs or 54 mins
Queueing Models (Henry C. Co) 49
Resource Utilization
service rate =
Queueing Models (Henry C. Co) 50
Resource Utilization
service rate ==
Lq =
Queueing Models (Henry C. Co) 51
Resource Utilization
Arrival Rate = 10.0= 1.0)
= 0.0= 0.0)
service rate ==
Lq =
Lq
Queueing Models (Henry C. Co) 52
Resource Utilization
Arrival Rate = 10.0= 1.0)
= 0.0= 0.0)
service rate ==
Lq =
Lq
Queueing Models (Henry C. Co) 53
Resource Utilization
Arrival Rate = 10.0= 1.0)
= 0.0= 0.0)
service rate ==
Lq =
Lq
Queueing Models (Henry C. Co) 54
Flexibility/Utilization Trade-off
Utilization = 1.0= 0.0
Queueing Models (Henry C. Co) 55
Flexibility/Utilization Trade-off
Utilization = 1.0= 0.0
L Lq
WWq
Queueing Models (Henry C. Co) 56
Flexibility/Utilization Trade-off
Utilization = 1.0= 0.0
L Lq
WWq
Queueing Models (Henry C. Co) 57
Flexibility/Utilization Trade-off
Utilization = 1.0= 0.0
L Lq
WWq
High utilizationLow flexibilityPoor service
Low utilizationHigh flexibilityGood service
Queueing Models (Henry C. Co) 58
Queues and Flexibility Low utilization levels ( < 0.6 ) provide
better service levels greater flexibility lower waiting costs (e.g., lost business)
High utilization levels ( > 0.9 ) provide better equipment and employee utilization fewer idle periods lower production/service costs
Must trade off benefits of high utilization levels with benefits of flexibility and service
Queueing Models (Henry C. Co) 59
Cost Trade-offs
Utilization = 1.0= 0.0
Cost
Queueing Models (Henry C. Co) 60
Cost Trade-offs
Utilization = 1.0= 0.0
Cost
Cost ofWaiting
Queueing Models (Henry C. Co) 61
Cost Trade-offs
Utilization = 1.0= 0.0
Cost
Cost ofWaiting
Cost ofService
Queueing Models (Henry C. Co) 62
Cost Trade-offs
Utilization = 1.0= 0.0
Cost
CombinedCombinedCostsCosts
Cost ofWaiting
Cost ofService
Queueing Models (Henry C. Co) 63
Queues and Simulation Only simple queues can be
mathematically analyzed “Real world” queues are often very
complex multiple servers, multiple queues balking, reneging, queue jumping machine breakdowns networks of queues, ...
Need to analyze, complex or not Computer simulation !
Queueing Models (Henry C. Co) 64
Adding an extra server Reduces the expected queue length and
waiting time greatly. Reduces the server’s utilization level
significantly. In some cases, a manager wants the
expected customer waiting time is below certain critical level. Otherwise, he may lose customers.
Questions
Queueing Models (Henry C. Co) 65