SlideShare a Scribd company logo
1 of 23
Download to read offline
QUEUING THEORY<br />Introduction<br />Queuing theory deals with problems that involve waiting (or queuing). It is quite common that instances of queue occurs everyday in our daily life. Examples of queues or long waiting lines might be<br />,[object Object]
Waiting for a train or bus.
Waiting at barber saloon.
Waiting at doctors’ clinic.Whenever a customer arrives at a service facility, some of them usually have to wait before they receive the desired service. This form a queue or waiting line and customer feel discomfort either mentally or physically because of long waiting queue.<br />             We infer that queues from because the service facilities are inadequate. If service facilities are increased, then the question arise how much to increase? For example, how many buses would be needed to avoid queues? How many reservation counters would be needed to reduce the queue? Increase in number of buses and reservation counters requires additional resources. At the same time, cost due to customer dissatisfaction must also be considered.<br />Symbols  and notations:<br />n = total number of customers in the system,  both waiting     and in service<br />µ = average number of customers being serviced per unit of time.<br />λ = average number of customers arriving per unit of time.<br />C = number of parallel service channels<br />Ls or E(n) = average number of customers in the system, both waiting in the service.<br />Lq  or E(m) = average number of customers waiting in the queue<br />Ws or E(w) = average wating time of a customer in the system both waiting and in service<br />Wq or E(w) = average waiting time of a customer in the queue<br />Pn (t = probability that there are n customer in the queue<br />total cost of the system<br />cost                               <br />cost of service<br />  cost of waiting<br /> optical service level    level of service<br />Queuing system<br />The customers arrive at service counter (single or in a group) and attended by one or more servers. A customer served leaves the system after getting the service. In general, a queuing system comprise with two components, the queue and the service facility. The queue is where the customers are waiting to be served. The service facility is customers being served and the individual service stations.<br /> SERVICE SYSTEM <br />The service is provided by a service facility (or facilities). This may be a person (a bank teller, a barber, a machine (elevator, gasoline pump), or a space (airport runway, parking lot, hospital bed), to mention just a few. A service facility may include one person or several people operating as a team. <br />There are two aspects of a service system—(a) the configuration of the service system and (b) the speed of the service.<br />                          <br />   Configuration of the service system <br />The customers’ entry into the service system depends upon the queue conditions. If at the time of customers’ arrival, the server is idle, then the customer is served immediately. Otherwise the customer is asked to join the queue, which can have several configurations. By configuration of the service system we mean how the service facilities exist. Service systems are usually classified in terms of their number of channels, or numbers of servers. <br />        Single Server – Single Queue<br /> <br />      The models that involve one queue – one service station facility are called single server models where customer waits till the service point is ready to take him for servicing. Students’ arriving at a library counter is an example of a single server facility.<br />       <br />Several (Parallel) Servers – Single Queue <br />            In this type of model there is more than one server and each server provides the same type<br />       of facility. The customers wait in a single queue until one of the service channels is ready to       take them in for servicing<br />                                                        <br />     <br />       <br />Several Servers – Several Queues<br /> <br />       This type of model consists of several servers where each of the servers has a different queue. Different cash counters in an electricity office where the customers can make payment in respect of their electricity bills provide an example of this type of model.<br />                                                                <br />      Service facilities in a series <br />       In this, a customer enters the first station and gets a portion of service and then moves on to the next station, gets some service and then again moves on to the next station. …. and so on, and finally leaves the system, having received the complete service. For example, machining of a certain steel item may consist of cutting, turning, knurling, drilling, grinding, and packaging operations, each of which is performed by a single server in a series. Service Facility.<br />Characteristics of Queuing System<br />In designing a good queuing system, it is necessary to have a good <br />Information about the model. The characteristic listed below would<br />Provide sufficient information. <br />,[object Object]
The service mechanism.
The queue discipline.
The number of service channels.
Number of Service Stages1. The Arrival pattern.<br />Arrivals can be measured as the arrival rate or the interarrival time<br />(time between arrivals).<br />Interarrival time =1/ arrival rate<br />These quantities may be deterministic or stochastic (given by a<br />propbability distribution).<br />Arrivals may also come in batches of multiple customers, which is<br />called batch or bulk arrivals. The batch size may be either deter-<br />ministic or stochastic.<br /> (i) Balking: The customer may decide not to enter the queue upon<br />Arrival, perhaps because it is too long.<br />(ii) Reneging: The customer may decide to leave the queue after<br />Waiting a certain time in it.<br />(iii) Jockeying: If there are multiple queues in parallel the customers<br />May switch between them.<br />(iv) Drop-o®s: Customers may be dropped from the queue for rea-<br />Sons outside of their control. (This can be viewed as a general-<br />Isation of reneging.)<br />2. Service Pattern<br />As with arrival patterns, service patterns may be deterministic or<br />stochastic. There may also be batched services.<br />The service rate may be state-dependent. (This is the analoge of<br />impatience with arrivals.)<br />Note that there is an important di®erence between arrivals and ser-<br />vices. Services do not occur when the queue is empty (i.e. in this<br />case it is a no-op).<br />3. Queue Discipline<br />This is the manner by which customers are selected for service.<br />,[object Object]
Last in First Out (LIFO), also called
Service in Random Order (SIRO).     (iv) Priority Schemes. Priority schemes are either:<br />Preemptive: A customer of higher priority immediately displaces<br />any customers of lower priority already in service. The displaced customer's service may be either resumed from where<br />it was left o®, or started a new.<br />Non-Preemptive: Customers with higher priority wait current<br />service completes, before being served.<br />4.The number of service channels<br />5. Number of Service Stages<br />Customers are served by multiple servers in series.<br />In general, a multistage queue may be a complex network with feed-<br />Back<br />Application of queuing  theory:<br />Queing theory has been applied to a great  variety of business situations. Here we shall discuss a few problem s where the theory may be applied-<br />,[object Object]
Waiting line theory can be used to analyze the delays  at the toll booths of bridges  and tunnels.
Waiting line theory can   be used  to improve the customers service at restaurants,cafeteria ,gasoline service station , airline counters,hospitals etc,
Waiting line theory can be used to determine  the proper determine the proper number docks to be constructed  in the building of terminal facilities  for trucks &ships.
Several manufacturing firms have attacked the problems of machine break down  &repairs  by utilizing this theory . Waiting line theory can be used to determine the number of personnal to be employed  so that thee cost  of the production loss from down time   & the cost f repairman  is minimized.
Queuing theory has been extended to study a wage incentive plan
Queuing theory (Limitations)
Most of the queuing models are quite complex & cannot be easily understood.
Many times  form of theoretical distribution  applicable to given queuing situations is not known.
If the queuing discipline  is not in” first in, first out”, the study of queuing problems become more difficult. <br />BASIC POINTS<br />Customer:> (Arrival)<br />The arrival unit that requires some services to performed.<br />Queue:>The number of Customer  waiting to be served. <br />Arrival Rate (λ):>The rate which customer arrive to the service station.<br />Service rate (µ) :> The rate at which the service unit can provide sevices    to  the customer<br />If Utilization Ratio Or Traffic intensity i.e  λ  /µ<br />λ  / µ > 1  Queue is growing without end.<br />λ  / µ < 1  Length of Queue is go on diminishing.<br />λ  /µ = 1  Queue length remain constant.<br />When Arrival Rate (λ) is less than Service rate (µ) the system is working .<br />i.e  λ< µ (system work)<br />Formulas<br />µ=Service Rate<br />λ= Arrival Rate<br />1. Traffic Intensity (P)=  λ  /µ<br />2. Probability Of System Is Ideal (P0) =1-P<br />                                                           P0 = 1- λ /µ<br />3. Expected Waiting Time In The System (Ws) = 1/  (µ- λ)<br />4. Expected Waiting Time In Quie   (Wq) = λ / µ(µ- λ)<br />5. Expected Number Of Customer In The System (Ls)= λ / µ(µ-λ)<br />     Ls=Length Of System<br />6. Expected Number Of Customers In The Quie  (Lq)= λ 2/ µ(µ- λ)<br />7. Expected Length Of Non-Empty Quie  (Lneq)= µ/ (µ- λ)<br />8. What Is The Probability Track That That K Or More Than K                                                      Customers In The System.<br />    P >=K        (P Is Greater Than Equal To K)<br />    = (λ  /µ)K<br />9. What Is The Probability That More Than K Customers Are In The               System (  P>K)= (λ  /µ)K+1<br />10. What Is The Probability That Atleast One Customer Is Standing In Quie.        P=K=(λ  /µ)2<br />11. What Is The Probability That Atleast Two Customer In The System<br />P=K=(λ  /µ)2<br />Solved Example.<br /> Question 1.People arrive at a cinema ticket booth in a poisson distributed arrival rate of 25per hour. Service rate is exponentially distributed with an average time of 2 per min.<br />Calculate the mean number in the waiting line, the mean waiting time , the mean number in the system , the mean time in the system and the utilization factor?<br />Solution:<br />Arrival rate  λ=25/hr<br />Service rate µ= 2/min=30/hr<br />Length of Queue (Lq)= λ 2/ µ(µ- λ)  <br />                                  = 252/(30(30-25))<br />                                 =4.17 persone<br /> Expected Waiting Time In Quie   (Wq) = λ / µ(µ- λ)<br />                                                                      =25/(30(30-25))<br />                                                                     =1/6 hr= 10 min<br />Expected Waiting Time In The System (Ws) = 1/  (µ- λ)<br />                                                                             =1/(30-25)<br />                                                                            =1/5hr= 12 min<br />Utilization Ratio = λ  /µ<br />                               =25/30<br />                              =0.8334 = 83.34%<br />Question 2. Assume that at a bank teller window the customer arrives at a average rate of 20 per hour according to poission distribution .Assume also that the bank teller spends an distributed customers who arrive from  an infinite population are served on  a first come first services basis and there is no limit to possible queue length.<br /> 1.what is the value of utilization factor?<br /> 2.What is the expected waiting time in the system per customer?<br /> 3.what is the probability of zero customer in the system?<br />Solution:<br />Arrival rate   λ=20 customer per hour<br />Service rate  µ= 30 customer per hour<br />     1.Utilization Ratio = λ  /µ <br />,[object Object],        2. Expected Waiting Time In The System (Ws) =  1/  (µ- λ)<br />                                                                               =1/(30-20)<br />                                                                               =1/10 hour = 6 min<br />3. Probability of zero customers in the system P0  = 1 – P<br />                                                                                         =1- 2/3 = 1/3<br />Question 3 : Abc company has one hob regrinding machine. The hobs needing grinding are sent from company’s tool crib to this machine which is operated one shift per day of 8 hours duration. It takes on the average half an hour to regrind a hob. The arrival of hobs is random with an average of 8 hobs per shift.<br />,[object Object]
What is average time for the hob to be in the regrinding section?
The management is prepared to recruit another grinding operator when the utilization of the machine increases to 80%. What should the arrival rate of hobs then be?Solution: : Let us calculate arrival rate and service rate per shift of 8 hours.<br />Arrival rate λ=8 shift<br />Service rate µ=8x60/30=16 /shift<br />,[object Object]
Pb =arrival rate/service rate=8/16=0.50=50%
Average time for the hob to be in the grinding section.  i.e.,   average time in the queue system=ws<br />ws =   1/( µ- λ)=1/16-8=1/8 shift=1/8x8=1 hour<br />,[object Object]

More Related Content

What's hot

Queueing Theory and its BusinessS Applications
Queueing Theory and its BusinessS ApplicationsQueueing Theory and its BusinessS Applications
Queueing Theory and its BusinessS ApplicationsBiswajit Bhattacharjee
 
Queuing theory
Queuing theoryQueuing theory
Queuing theoryAbu Bashar
 
Introduction to queueing theory
Introduction to queueing theoryIntroduction to queueing theory
Introduction to queueing theoryPresi
 
Queuing Theory - Operation Research
Queuing Theory - Operation ResearchQueuing Theory - Operation Research
Queuing Theory - Operation ResearchManmohan Anand
 
Operational research queuing theory
Operational research queuing theoryOperational research queuing theory
Operational research queuing theoryVidhya Kannan
 
Case Study of a Pizza Restaurant using House of Quality or Quality Function D...
Case Study of a Pizza Restaurant using House of Quality or Quality Function D...Case Study of a Pizza Restaurant using House of Quality or Quality Function D...
Case Study of a Pizza Restaurant using House of Quality or Quality Function D...Deepanshu Saini
 
Queuing theory and simulation (MSOR)
Queuing theory and simulation (MSOR)Queuing theory and simulation (MSOR)
Queuing theory and simulation (MSOR)snket
 
Operating characteristics ofqueuing system
Operating characteristics ofqueuing systemOperating characteristics ofqueuing system
Operating characteristics ofqueuing systemAminul Tanvin
 
Aggregate planning
Aggregate planningAggregate planning
Aggregate planningAtif Ghayas
 
Facility location of services
Facility location of servicesFacility location of services
Facility location of servicesSnigdha Vashishth
 
Facility location and techniques
Facility location and techniquesFacility location and techniques
Facility location and techniquesPiyush Sharma
 
queueing problems in banking
queueing problems in bankingqueueing problems in banking
queueing problems in bankingMani Deep
 
Waiting line
Waiting lineWaiting line
Waiting lineE P John
 
Waiting Lines & Simulation
Waiting Lines & SimulationWaiting Lines & Simulation
Waiting Lines & SimulationNavitha Pereira
 

What's hot (20)

Queueing Theory and its BusinessS Applications
Queueing Theory and its BusinessS ApplicationsQueueing Theory and its BusinessS Applications
Queueing Theory and its BusinessS Applications
 
Queuing theory
Queuing theoryQueuing theory
Queuing theory
 
Introduction to queueing theory
Introduction to queueing theoryIntroduction to queueing theory
Introduction to queueing theory
 
Queuing Theory - Operation Research
Queuing Theory - Operation ResearchQueuing Theory - Operation Research
Queuing Theory - Operation Research
 
Operational research queuing theory
Operational research queuing theoryOperational research queuing theory
Operational research queuing theory
 
Case Study of a Pizza Restaurant using House of Quality or Quality Function D...
Case Study of a Pizza Restaurant using House of Quality or Quality Function D...Case Study of a Pizza Restaurant using House of Quality or Quality Function D...
Case Study of a Pizza Restaurant using House of Quality or Quality Function D...
 
Queuing theory and simulation (MSOR)
Queuing theory and simulation (MSOR)Queuing theory and simulation (MSOR)
Queuing theory and simulation (MSOR)
 
Queuing theory
Queuing theoryQueuing theory
Queuing theory
 
Operating characteristics ofqueuing system
Operating characteristics ofqueuing systemOperating characteristics ofqueuing system
Operating characteristics ofqueuing system
 
Goal programming
Goal programmingGoal programming
Goal programming
 
Aggregate planning
Aggregate planningAggregate planning
Aggregate planning
 
Facility location of services
Facility location of servicesFacility location of services
Facility location of services
 
Facility location and techniques
Facility location and techniquesFacility location and techniques
Facility location and techniques
 
Transportation problems
Transportation problemsTransportation problems
Transportation problems
 
Ramniwas final
Ramniwas finalRamniwas final
Ramniwas final
 
unbalanced transportation problem
unbalanced transportation problemunbalanced transportation problem
unbalanced transportation problem
 
queueing problems in banking
queueing problems in bankingqueueing problems in banking
queueing problems in banking
 
Queueing theory
Queueing theoryQueueing theory
Queueing theory
 
Waiting line
Waiting lineWaiting line
Waiting line
 
Waiting Lines & Simulation
Waiting Lines & SimulationWaiting Lines & Simulation
Waiting Lines & Simulation
 

Viewers also liked

Simulasi - Pertemuan III
Simulasi - Pertemuan IIISimulasi - Pertemuan III
Simulasi - Pertemuan IIIDimara Hakim
 
Simulation & Modeling - Smilulation Queuing System
Simulation & Modeling - Smilulation Queuing SystemSimulation & Modeling - Smilulation Queuing System
Simulation & Modeling - Smilulation Queuing SystemMaruf Rion
 
Chp. 2 simulation examples
Chp. 2 simulation examplesChp. 2 simulation examples
Chp. 2 simulation examplesPravesh Negi
 
Inventory Management
Inventory ManagementInventory Management
Inventory Managementanoos
 
Inventory management
Inventory managementInventory management
Inventory managementKuldeep Uttam
 

Viewers also liked (6)

Simulasi - Pertemuan III
Simulasi - Pertemuan IIISimulasi - Pertemuan III
Simulasi - Pertemuan III
 
Simulation & Modeling - Smilulation Queuing System
Simulation & Modeling - Smilulation Queuing SystemSimulation & Modeling - Smilulation Queuing System
Simulation & Modeling - Smilulation Queuing System
 
Chp. 2 simulation examples
Chp. 2 simulation examplesChp. 2 simulation examples
Chp. 2 simulation examples
 
Inventory Management
Inventory ManagementInventory Management
Inventory Management
 
Inventory management
Inventory managementInventory management
Inventory management
 
Inventory Management
Inventory ManagementInventory Management
Inventory Management
 

Similar to Queuing theory

Queuing theory
Queuing theoryQueuing theory
Queuing theoryAbu Bashar
 
chapter11 liner model programing.pptx
chapter11 liner model programing.pptxchapter11 liner model programing.pptx
chapter11 liner model programing.pptxJAMESFRANCISGOSE
 
Queuing theory network
Queuing theory networkQueuing theory network
Queuing theory networkAmit Dahal
 
Simulation chapter 4
Simulation chapter 4Simulation chapter 4
Simulation chapter 4Hagos Guesh
 
Queuing theory and its applications
Queuing theory and its applicationsQueuing theory and its applications
Queuing theory and its applicationsDebasisMohanty37
 
Theory of queuing systems
Theory of queuing systemsTheory of queuing systems
Theory of queuing systemsKaushik Kundu
 
International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI) International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
 
4-Queuing-System-ioenotes.pdf
4-Queuing-System-ioenotes.pdf4-Queuing-System-ioenotes.pdf
4-Queuing-System-ioenotes.pdfubaidullah75790
 
Service operation management 7
Service operation management 7Service operation management 7
Service operation management 7Judianto Nugroho
 
Waiting Line Management
Waiting Line ManagementWaiting Line Management
Waiting Line ManagementRaman Kumar
 
Queueing theory
Queueing theoryQueueing theory
Queueing theoryDevan P.D
 
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...IOSR Journals
 
Decision Sciences_SBS_9.pdf
Decision Sciences_SBS_9.pdfDecision Sciences_SBS_9.pdf
Decision Sciences_SBS_9.pdfKhushbooJoshiSBS
 

Similar to Queuing theory (20)

Queuing theory
Queuing theoryQueuing theory
Queuing theory
 
Waiting line theroy
Waiting line theroyWaiting line theroy
Waiting line theroy
 
4Queuing_Theory.ppt
4Queuing_Theory.ppt4Queuing_Theory.ppt
4Queuing_Theory.ppt
 
chapter11 liner model programing.pptx
chapter11 liner model programing.pptxchapter11 liner model programing.pptx
chapter11 liner model programing.pptx
 
Queuing theory network
Queuing theory networkQueuing theory network
Queuing theory network
 
Queuing Theory by Dr. B. J. Mohite
Queuing Theory by Dr. B. J. MohiteQueuing Theory by Dr. B. J. Mohite
Queuing Theory by Dr. B. J. Mohite
 
Simulation chapter 4
Simulation chapter 4Simulation chapter 4
Simulation chapter 4
 
Queuing theory and its applications
Queuing theory and its applicationsQueuing theory and its applications
Queuing theory and its applications
 
QUEUING THEORY
QUEUING THEORY QUEUING THEORY
QUEUING THEORY
 
Theory of queuing systems
Theory of queuing systemsTheory of queuing systems
Theory of queuing systems
 
Waiting Lines.pptx
Waiting Lines.pptxWaiting Lines.pptx
Waiting Lines.pptx
 
International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI) International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI)
 
Theory of queues
Theory of queuesTheory of queues
Theory of queues
 
4-Queuing-System-ioenotes.pdf
4-Queuing-System-ioenotes.pdf4-Queuing-System-ioenotes.pdf
4-Queuing-System-ioenotes.pdf
 
Service operation management 7
Service operation management 7Service operation management 7
Service operation management 7
 
Waiting Line Management
Waiting Line ManagementWaiting Line Management
Waiting Line Management
 
Queueing theory
Queueing theoryQueueing theory
Queueing theory
 
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...
 
Decision Sciences_SBS_9.pdf
Decision Sciences_SBS_9.pdfDecision Sciences_SBS_9.pdf
Decision Sciences_SBS_9.pdf
 
Queuing theory
Queuing theoryQueuing theory
Queuing theory
 

Queuing theory