The statistical pattern of the arrival can be indicated through the probability distribution of the number of the arrivals in an interval.

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Dina Phelps
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1 Itroductio Queuig are the most freuetly ecoutered roblems i everyday life. For examle, ueue at a cafeteria, library, bak, etc. Commo to all of these cases are the arrivals of objects reuirig service ad the attedat delays whe the service mechaism is busy. Waitig lies caot be elimiated comletely, but suitable techiues ca be used to reduce the waitig time of a object i the system. A log waitig lie may result i loss of s to a orgaizatio. Waitig time ca be reduced by rovidig additioal service facilities, but it may result i a icrease i the idle time of the service mechaism. Basic Termiology Queuig Model It is a suitable model used to rereset a service orieted rob lem, where s arrive radomly to receive some service, the service time beig also a radom variable. Arrival The statistical atter of the arrival ca be idicated through the robability distributio of the umber of the arrivals i a iterval. ervice Time The time take by a server to comlete service is kow as service time. erver It is a mechaism through which service is offered. Queue Discilie It is the order i which the members of the ueue are offered service. i.e, It is the rule accordigly to which s are selected for service whe ueue has bee formed. The most commo discilies are. First come First servicesfcf. First i First OutFIFO. Last i First outlifo 4. electio for service i Radom orderiro Poisso Process It is a robabilistic heomeo where the umber of arrivals i a iterval of legth t follows a Poisso distributio with arameter t, where is the rate of arrival. Queue Waitig lies A grou of items waitig to receive service, icludig those receivig the service, is kow as ueue. Waitig time i ueue W Time set by a i the ueue before beig served. Waitig time i the system W D.J Page

2 It is the total time set by a i the system. It ca be calculated as follows: Waitig time i the system = Waitig time i ueue + ervice time Queue legth L ersos i the system at ay time Average legth of lie The umber of s i the ueue er uit of time Average idle time The average time for which the system remais idle Bulk Arrivals If more tha oe eters the system at a arrival evet, it is kow as bulk arrivals. Note that bulk arrivals are ot embodied i the models of the subseuet sectios. Queuig ystem A ueuig system ca be comletely described by The iut or arrival Patte The service mechaism or service atter The ueuig discilie Customer s behavior. Geeral form of Queuig Model The geeral form of a ueuig model as a / b/ c: d / e. a =Arrival Distributio. b =. c = 4. d = Differet tyes of Queuig Model. M/M/: /FIFO system. M/M/C: /FIFO system. M/M/: /FIFO system 4. M/M/: /FIFO system D.J Page

3 Oeratig characteristic of a Queuig ystem Queuig legth L ystem legthl Waitig time i ueue W Waitig time i system W. Traffic itesity/ Utilizatio Factor It is the ratio of average arrival Where there are oe server rate ad average service rate Where there are multi server rate ad average service rate Mea arrival rate Mea service rate Mea arrival rate Mea service rate C Alicatio of Queuig Model. Busiess situatios deart metal stores, ciema halls, etrol ums, atiets cliic, airlies couters etc.. chedulig of jobs i roductio cotrol.. olutio of ivetory cotrol. The M/M/: /FIFO system This is a ueuig model i which the arrival is Marcovia ad dearture distributio is also Marcovia, umber of server is oe ad size of the ueue is also Marcovia, o. of server is oe ad size of the ueue is ifiite ad service discilie is st come st serve FCF ad the callig source is also fiite. Assumtios ad Notatios. = umber of s i system. µ=mea service rate. =mea arrival rate 4. P t=robability of s i system at time t 5. Probability of oe arrival i the system durig t = t + Oh 6. Probability of more tha oe arrival i the system durig t = Oh 7. Probability of o arrival i the system durig t =- t + Oh 8. Probability of oe beig service i time t = µ t + Oh 9. Probability of more tha oe beig service i time t = Oh. Probability of ot a sigle beig service i time t =- µ t + Oh Let t+ tbe the robability of s i the system at the time t + t For > Evet time t arrivals i dearture i time t + D.J Page

4 4 + - t t t t or, or, t t t t t o t t o t t Takig the limit, we get the followig differetial euatio, d t t t. For = Evet time t arrivals i dearture i --- time t + t t t o Or, t t t t o Or, t t o t. Takig the limit, we get the followig differetial euatio, d t t, whe =. For steady state coditios ' t d t Therefore, the above reduces to differetial euatios,, where >, where =. We have. Puttig =,,, we get D.J Page 4

5 .., where >, Hece Characteristics L. Exected Legth of Queue i i i L L [ ] L L. Exected Legth of ystem: i i i i... Therefore, L D.J Page 5

6 . Waitig time i ystem: W L 4. Variace of Queue legth: W W Workig Formulae. Probability of zero uits i the ueue P. Average ueue legth L. Average umber of uits i the system L 4. Average waitig time of a arrival W 5. Average waitig time of a arrival i the system W Examle : A Televisio reairma fids that the time set o his jobs has a exoetial distributio with mea miuets. If he reairs the sets i the order i which they come i, ad if the arrivals of sets are aroximately Poisso with a average rate of er 8 hours day which is the reairs ma idle time each day?fid the exected umber of uits i the system ad i the ueue? olutio: it is a M\M\: \FCF ueuig system. Where, = mea arrival rate =/8 uits er hour µ=mea service rate =mis er hour. Therefore = /µ=/8.=5/8. Exected umber of uits i the system. Exected umber of uits i the ueue L 5 sets 5 L = L =. Probability of reairma beig idle =robability of havig o T.V sets i the system 5 = = Therefore reairma will remai idle for 8 hours er day. Exercises. What do you uderstad by a ueue? Give some imortat alicatios of ueuig theory. 8 D.J Page 6

7 . A barber with a oe- ma sho takes exactly miutes to comlete oe haircut. If s arrive accordig to a Poisso rocess at a rate of oe every 4 miutes, how log o the average must a wait for service?. a Defie a ueue. What are the basic characteristics of a ueuig system? b Prove that c Prove that s L L of / M /: / FCF M ueuig model. WBUT CH-8 L of M / M /: / FCF ueuig model 4. For a M / M /: / FCF ueuig model, derive the exressios for: WBUT a The steady state euatio. b Exected umber of s i the system. c Exected umber of s i the ueue. 5. A two chael waitig lie with Poisso arrival has a mea arrival rate of 5 er hour ad exoetial service with a mea service rate of 75 er hour for each chael. Fid i ii The robability of the emty system. The robability that a arrival i the system will have to wait. Preared by Dr. Diak Kumar Jaa D.J Page 7

First come, first served (FCFS) Batch

First come, first served (FCFS) Batch Queuig Theory Prelimiaries A flow of customers comig towards the service facility forms a queue o accout of lack of capacity to serve them all at a time. RK Jaa Some Examples: Persos waitig at doctor s