M(t)/M/1 Queueing System with Sinusoidal Arrival Rate

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1 20 TUTA/IOE/PCU Journal of he Insiue of Engineering, 205, (): TUTA/IOE/PCU Prined in Nepal M()/M/ Queueing Sysem wih Sinusoidal Arrival Rae A.P. Pan, R.P. Ghimire 2 Deparmen of Mahemaics, Tri-Chandra Campus, Ghanaghar, TU, Kahmandu, Nepal 2 Deparmen of Naural Sciences (Mahemaics), Kahmandu Universiy, Kavre, Nepal Corresponding auhor: panananda88@gmail.com Absrac: This paper deals wih he sudy of M ()/M/ queueing sysem wih cusomers arrive o he sysem wih sinusoidal arrival rae funcion λ () and are served exponenially wih he rae. On formulaing he mahemaical model, we obain he expressions for mean waiing ime in he queue, mean ime spen in he sysem, mean number of cusomers in he queue and in he sysem by using recursive mehod. Some numerical illusraions are also obained by using compuing sofware so as o show he applicabiliy of he model under sudy. Keywords: Queue, Sinusoidal funcion, Exponenial, Probabiliy. Inroducion From very beginning of inroducion of queueing sysem, several auhors sudied M/M/ queueing model in various provisions and frameworks. Of course he M/M/ model has been sudied exensively and much is known abou is ime - dependen or ransien behavior. Some of he pioneering and laes works are noable. Abae and Whi [] described he evoluion of he classical M/M/ queue. They obained simple approximaions and srucural heorems ha expose he essenial naure of he ransien behavior. Abae and Whi [2] developed a beer undersanding of he ransien behavior of queues and relaed sochasic flow sysems so ha we can provide relaively simple descripions suiable for engineering purposes. Baccelli and Massey [3] presened a derivaion of he ransien disribuion for he queue lengh and he busy period of he M/M/ queue ha follows purely from he sample pah behavior of he process. Li e al. [6] sudied M/M/ queues in which he service raes depend on he number of he cusomers served since he beginning of he curren busy period. They presened a simple and compuaionally racable scheme which recursively compues he saionary probabiliies of he queue lengh. Kinaeder and Lee [5] provided a new approach o he compuaion of he Laplace ransform of he lengh of he busy period of he M/M/ queue wih consrained workload (finie dam) wihou he use of complex analysis. Choudhury and Borhakur [4] deal wih saisical inference wih regard o he classical single server Markovian queueing model. Ghimire and Ghimire [7] deal wih he sudy of M/M/ queue wih heerogeneous arrival and deparure wih he provision of server vacaions and breakdowns cusomer arrive service faciliies wih poison process and exponenial service ime disribuion. Murhy e al. [9] sudied a generalized queueing model in

2 Pan and Ghimire 2 which cusomers are served as a bach of size k a a ime excep when here are less han k cusomers in he sysem a he ime of service. For developing hese independen models wih bulk service rule, hey made use of he dependence srucure given by Rao K.S (986). Maurya [7] explored he probabiliy generaing funcions using Rouche s heorem in boh cases of slower and faser arrival raes of he queueing model ino consideraion. Jindal and Sharma [3] sudied M/M/ queue under non preempive service prioriy discipline. Rasouli e al. [20] proposed a new analyical model o esimae he energy consumpion in clusered WSNS using M/M/ queueing model for all sensor nodes and developed an analyical model for energy saving by reducing he number of ransiions beween idle sae and acive sae in all sensor nodes. Recenly, Kalidass and Ramanah [4] explained explici expressions for he ime dependen probabiliies of he M/M/ queue wih server vacaions under a muliple vacaion scheme. They also obained he dependen performance measures of he sysem. Ibe and Isijola [] deal wih an M/M/ queueing sysem in which wo ypes of vacaions can be aken by he server. Maurya [8] demonsraed a mahemaical modeling for analyzing a Markovian queueing sysem wih wo heerogeneous servers and working vacaion and obained various performance measures of he Markovian queueing sysem wih varying parameers under seady sae using marix geomeric mehod. M/M/ queueing model sudied by aforemenioned several auhors could no address some real queueing problems where in arrival rae funcion is piecewise coninuous and occurs in real life siuaions in inernaional border checking process where arrival rae funcion is characerized by sinusoidal funcion λ () = ƛ sin () in he case when arrival rae rapidly flucuaes, where is he delay probabiliy and is he service qualiy parameer ha = ƛ for posiive consans ƛ, 0< < and =2/, Ψ is he cycle lengh or period.sinusoidal arrival rae funcion is applicable when we impose he flow conrol policy in which cusomer is expeced o arrive wihin an agreed appoinmen ime window insead of a a specific appoinmen ime. Very rare lieraures can be found in which he sinusoidal arrival rae has been employed in he sudy. So i is worhwhile o menion some of he works done on he line. Jagerman [2] discussed he ime behavior of blocking in a fully available N-runk group whose rae parameer iself was considered o vary wih ime. Rohkopf and Oren [2] sudied effecive compuaional mehods for dealing wih queues having non-saionary arrival processes. Heyman and Whi [0] deal he asympoic behavior of he M /G/c queue having a Poisson arrival process wih a general deerminisic inensiy. Green e al. [9] sudied a beer undersanding of how non-saionary affecs delays in queueing sysems. Green and Kolesar [8] discussed an easy - o - compue approximaion for deermining long run average performance measures for muli server Markovian queues wih periodic arrival raes. Eick e al. [6] developed a beer undersanding of he ime-dependen behavior in M /G/ queueing model. Dong and Whi [5] invesigaed he consequences of fiing a birh and deah (BD) process o a muli-server queue wih a periodic ime varying arrival rae funcion o beer undersanding of sysem. In he presen sudy, we give some insighs ino M ()/M/ queueing model under he provision of sinusoidal arrival rae funcion and obain various performance measures such as expeced number of cusomers in he sysem, expeced number of cusomers in he queue, expeced ime

3 22 M()/M/ Queueing Sysem wih Sinusoidal Arrival Rae per cusomer in he sysem, expeced waiing ime per cusomer in he queue, raffic inensiy. To show ha our model sudied is pracically applicable we show he numerical resuls for various parameers change. Our model can be applied in he pracical field where here is only one server and arrival of he cusomers in he sysem is periodic. I has ubiquious applicaions in he world where he securiy has become he grea issue. 2. Mahemaical Model The noaions used for our model are as follows: n = number of cusomers in he sysem a ime = arrival rae funcion ƛ = mean arrival rae µ = mean service rae P n = seady sae probabiliy of exacly n cusomers in he sysem P n () = ransien sae probabiliy of exacly n cusomers in he sysem a ime, assuming he sysem sared is operaion a ime zero. L s = expeced number of cusomers in he sysem L q = expeced number of cusomers in he queue. W s = expeced ime a cusomer spends in he sysem. W q = expeced waiing ime per cusomer in he queue. = raffic inensiy. The ransiion diagram for our model is shown in figure n- n n+ µ µ µ µ µ µ Fig. : Transiion diagram Wih he help of above ransiion diagram, he ransiion differenial equaions are: ), 0 2 In seady sae service sysem, when, 0, 0 0 3) 0 4 Solving (3) and (4) recursively, we have

4 Pan and Ghimire 23., 0 5 Now, nex sep is o find P 0 by using normalizing condiion ha yields Hence, 3. Performance measures of he sysem 3. Expeced number of cusomers in he sysem = 2 3 = Expeced number of cusomers in he queue L q = expeced number of cusomer in he sysem-expeced number in service (single server) = =. 3.3 Expeced ime per cusomer in he sysem (8) 9

5 24 M()/M/ Queueing Sysem wih Sinusoidal Arrival Rae. 3.4 Expeced waiing ime per cusomer in he queue 3.5 Traffic inensiy. (0) 2 4. Resuls and Discussion γ = γ= Ls Fig.2:Traffic inensiy vs ime γ= γ= Lq Ws Fig.3:Average number of cusomers in he sysem vs ime Fig.4:Average number of cusomers waiing in queue vs ime Fig.5:Average ime spen in he sysem vs ime

6 Pan and Ghimire γ= µ =20 µ = Wq 4.4 Ls Fig.6:Average waiing ime in queue vs ime Fig.7:Average number of cusomers in he sysem vs raffic inensiy µ=20 µ = µ =20 µ= Ws 0.2 Lq Fig.8:Average number of cusomers waiing in queue vs raffic inensiy Fig.9:Average ime spen in he sysem vs raffic inensiy mu =20 µ = Wq Fig.0:Average waiing ime in queue vs raffic inensiy

7 26 M()/M/ Queueing Sysem wih Sinusoidal Arrival Rae Fig. (2) Explores ha smaller he value of relaive frequency, he longer he ime ha he sysem ake o achieve higher value of raffic inensiy. This implies ha longer he ime o have cusomers in he queue. Fig. (3) predics ha smaller he value of, longer is he cycle ime and slower rae o accumulae he cusomers in he sysem as well as in he queue. Fig. (4) displays ha cusomers are waiing in queue from beginning. Bu as ime passes on, cusomers in queue increases faser when = 0.75 han =, which is realisic. Fig. (5) shows ha cycle ime spen in he sysem increases faser when we ake = 0.75 han =. This is also pracical in real life siuaion. Fig. (6) explains ha in he beginning of ime, cusomers have already waied and as he ime increase, waiing ime decreases sharply for a while and again repeas he process. Cycle ime is slower when = han = This can be experienced in he border crossing area in he check poin ec. Fig. (7) shows ha for fixed, iniially he server is less busy when service rae is 25 bu when service rae decreases o 20, he server is more busy. When numbers of cusomers in he sysem reach higher value, he raffic inensiy increases fas, his is pracically rue. Fig.(8) predics ha when service rae is 25, server is busy from beginning bu when 20, server is more busy afer someimes of he saring of he sysem. Bu for boh value of waiing number of cusomers gradually increases simulaneously as he busy facor increases. Fig. (9) elaboraes ha lesser is he service rae, more is he ime spen in he sysem. When service rae = 25, cusomer has no o wai in he beginning. Bu aferwards hey have o wai more han he cusomers when he service rae = 20, which is inherenly rue. Fig. (0) shows ha when service rae = 25, cusomers have already waied in he queue bu his is no so in he case when service rae is 20. Waiing ime is higher when service rae is higher. Lesser he service rae, below is he graph. Higher he service rae, above is he graph. Bu his is for a cerain values of uilizaion facor in a cycle. 5. Conclusion We have obained various performance measures such as expeced number of cusomers in he sysem, expeced number of cusomers in he queue, expeced ime per cusomer in he sysem, expeced waiing ime per cusomer in he queue, raffic inensiy. To show ha our model sudied is pracically applicable we show he numerical resuls for various parameers change. Our model can be applied in he pracical field where here is only one server and arrival of he cusomers in he sysem is periodic. I has ubiquious applicaions in he world where he securiy has become he grea issue. Such model was applied very well no only in modern age bu also in he ancien days knowingly or unknowingly such as firs come firs served service discipline has been well implemened in RAMAYANA and MAHABHARAT. Our model can be sudied as a furher research under he provision of sinusoidal service raes wih ransien consideraion. References [] Abae J and Whi W (987), Transien Behavior of he M/M/ Queue: Saring a he Origin. Queueing Sysems 2 : [2] Abae J and Whi W (988), Transien Behavior of he M/M/ Queue via Laplace ransforms. Applied probabiliy Trus 20 :

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