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WAITING LINES and QUEUING
THEORY MODELS
Isaac Lawrence Yvan C. Andaya
INTRODUCTION
• The Foundation of modern
queuing theory is based on
studies about automatic
dialing equipment made in
the e...
QUEUING THEORY
• The Planning and Analysis of service capacity.
• It is the study of Waiting Lines.
• Three basic componen...
Applications of Queuing Theory
• Telecommunications
• Traffic Control
• Determining the sequence of computer
operations
• ...
• Waiting lines tends to form even though a
system is basically under loaded. For
example: a fast food restaurant may have...
• In real situation, customers arrive at
random intervals rather than at evenly
spaced intervals, also some orders take
lo...
Implication of Waiting Lines
There are Reasons why managers have to be concerned
with waiting lines:
1.The cost of providi...
WAITING LINE COSTS
• The Main goal of waiting line analysis
is to minimize costs. There are two
type of cost in a queuing ...
Capacity costs
• Is defined as the costs of maintaining
the ability to provide service. Capacity
is lost when a service fa...
• Meanwhile the cost of customer waiting for
service could include: the salaries paid to
employees while they wait, the co...
• Queuing costs and service level
*
Optimal
Service
Level
Cost
Service Level
Cost of Providing Service
Total Expected Cost...
Characteristics of a Queuing
System
• There are three parts to a queuing system
1. The arrivals or inputs to the system (s...
Arrival Characteristics
• Arrival Characteristics have three major
characteristics, sizesize, patternpattern, and behavior...
• The Poisson distribution is
4,...3,2,1,0,for ==
−
X
X
e
XP
X
!
)(
λλ
P(X) = probability of X arrivals
X = number of arri...
!
)(
X
e
XP
X
λλ−
=
%.
)(.
!
)( 1413530
1
113530
0
2
0
02
====
−
e
P
%.
)(.
!
)( 2727060
1
213530
1
2
1
2
1
212
=====
−−
e...
Behavior of arrivals
– Join the Queue,Join the Queue, Most queuing models assume
customers are patient and will wait in th...
– Waiting lines can be either limitedlimited or unlimitedunlimited
– Queue discipline refers to the rule by which
customer...
Service Facility Characteristics
• Service systems are classified in terms of the number of
channels, or servers, and the ...
Service Time Distribution
– Service patterns can be either constant or random
– Constant service times are often machine c...
Performance Measure of
Queuing systems
Managers observed these five factors when evaluating
existing or proposed service s...
SINGLE CHANNEL EXPONENTIAL
SERVICE TIME MODEL
• This model involves a system that has one
server and there is no limit on ...
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Opersea report waiting lines and queuing theory

  1. 1. WAITING LINES and QUEUING THEORY MODELS Isaac Lawrence Yvan C. Andaya
  2. 2. INTRODUCTION • The Foundation of modern queuing theory is based on studies about automatic dialing equipment made in the early part of the 20th century by Danish telephone engineering “A.K. ERLANG”.
  3. 3. QUEUING THEORY • The Planning and Analysis of service capacity. • It is the study of Waiting Lines. • Three basic components of a queuing process are arrivals, service facilities and the actual waiting line. • Queuing theory has been used for operations research, manufacturing and system analysis.
  4. 4. Applications of Queuing Theory • Telecommunications • Traffic Control • Determining the sequence of computer operations • Airport traffic, airline ticket sales • Predicting computer performance
  5. 5. • Waiting lines tends to form even though a system is basically under loaded. For example: a fast food restaurant may have the capacity to handle an average of 100 orders/hour but may experience waiting lines even though the average number of orders is only 75/hour. The operative word is “AVERAGE”.
  6. 6. • In real situation, customers arrive at random intervals rather than at evenly spaced intervals, also some orders take longer to fill than the others. In short both arrivals and service time exhibit a high degree of variability, w/c results to temporary overloading – waiting lines.
  7. 7. Implication of Waiting Lines There are Reasons why managers have to be concerned with waiting lines: 1.The cost of providing a waiting space. 2.Should customers leave the line before being served or refuse to wait at all, the possible business loss should be considered. 3.A possible loss of goodwill. 4.Probable reduction in customer satisfaction. 5.The resulting congestion may interrupt other business operations and or customers.
  8. 8. WAITING LINE COSTS • The Main goal of waiting line analysis is to minimize costs. There are two type of cost in a queuing scenario; those costs relating to customers waiting for service and those relating to capacity.
  9. 9. Capacity costs • Is defined as the costs of maintaining the ability to provide service. Capacity is lost when a service facility is idle, since capacity cannot be stored.
  10. 10. • Meanwhile the cost of customer waiting for service could include: the salaries paid to employees while they wait, the cost of space for waiting and any loss of business due to customers refusing to wait and or going elsewhere. This loss of business is oftentimes called as ”Opportunity Loss.”
  11. 11. • Queuing costs and service level * Optimal Service Level Cost Service Level Cost of Providing Service Total Expected Cost Cost of Waiting Time
  12. 12. Characteristics of a Queuing System • There are three parts to a queuing system 1. The arrivals or inputs to the system (sometimes referred to as the calling populationcalling population) 2. The queue or waiting line itself 3. The service facility • These components have their own characteristics that must be examined before mathematical models can be developed
  13. 13. Arrival Characteristics • Arrival Characteristics have three major characteristics, sizesize, patternpattern, and behaviorbehavior – Size of the calling population • Can be either unlimited (essentially infiniteinfinite) or limited (finitefinite) – Pattern of arrivals • Can arrive according to a known pattern or can arrive randomlyrandomly • Random arrivals generally follow a PoissonPoisson distributiondistribution
  14. 14. • The Poisson distribution is 4,...3,2,1,0,for == − X X e XP X ! )( λλ P(X) = probability of X arrivals X = number of arrivals per unit of time λ = average arrival rate e = 2.7183 POISSON DISTRIBUTION
  15. 15. ! )( X e XP X λλ− = %. )(. ! )( 1413530 1 113530 0 2 0 02 ==== − e P %. )(. ! )( 2727060 1 213530 1 2 1 2 1 212 ===== −− ee P %. )(. )(! )( 2727060 2 413530 12 4 2 2 2 222 ===== −− ee P POISSON DISTRIBUTION
  16. 16. Behavior of arrivals – Join the Queue,Join the Queue, Most queuing models assume customers are patient and will wait in the queue until they are served and do not switch lines – BalkingBalking refers to customers who refuse to join the queue – RenegingReneging customers enter the queue but become impatient and leave without receiving their service – That these behaviors exist is a strong argument for the use of queuing theory to managing waiting lines
  17. 17. – Waiting lines can be either limitedlimited or unlimitedunlimited – Queue discipline refers to the rule by which customers in the line receive service – The most common rule is first-in, first-outfirst-in, first-out (FIFOFIFO) – Other rules are possible and may be based on other important characteristics – Other rules can be applied to select which customers enter which queue, but may apply FIFO once they are in the queue Waiting Line Characteristics
  18. 18. Service Facility Characteristics • Service systems are classified in terms of the number of channels, or servers, and the number of phases, or service stops • A single-channel systemsingle-channel system with one server is quite common • MultichannelMultichannel systemssystems exist when multiple servers are fed by one common waiting line • In a single-phase systemsingle-phase system the customer receives service form just one server • If a customer has to go through more than one server, it is a multiphase systemmultiphase system
  19. 19. Service Time Distribution – Service patterns can be either constant or random – Constant service times are often machine controlled – More often, service times are randomly distributed according to a negative exponential probabilitynegative exponential probability distributiondistribution – Models are based on the assumption of particular probability distributions – Analysts should take to ensure observations fit the assumed distributions when applying these models
  20. 20. Performance Measure of Queuing systems Managers observed these five factors when evaluating existing or proposed service systems. •The Average number of customers waiting, either in line or in the system. •The Average time customers wait or spend in the cue, either in line or in the system. •Average length of the cue. •System Utilization – refers to the percentage of capacity utilized. •The implied cost of a given level of capacity and its related waiting line. •The probability that an arrival will have to wait for service. •Probability that the service will be idle.
  21. 21. SINGLE CHANNEL EXPONENTIAL SERVICE TIME MODEL • This model involves a system that has one server and there is no limit on length of queue. 1.Queue discipline: FIFO 2.No Balking or Regening 3.Independent arrivals 4.Arrivals: Poisson distributed 5.Service times: negative exponential 6.Average service rate . Average arrival rate

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