An M/M/1/N Queueing Model with Retention of Reneged Customers and Balking

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1 America Joural of Operatioal Reearch, (): -5 DOI:.593/j.ajor.. A M/M// Queueig Model with Retetio of Reeged Cutomer ad Balkig Rakeh Kumar *, Sumeet Kumar Sharma School of Mathematic, Shri Mata Vaiho Devi Uiverity, Katra, Sub Pot- Office, Uiverity Campu, Potcode 83, Jammu ad Kahmir, Idia Abtract The cocept of cutomer balkig ad reegig ha bee exploited to a great extet i recet pat by the queuig modeler. Ecoomically, if we ee, the cutomer impatiece (due to balkig ad reegig) lead to the lo of potetial cutomer ad thereby reult ito the lo i the total reveue. Takig ito coideratio thi cutomer lo due to impatiece, a ew queuig model ha bee developed that deal with retetio of reeged cutomer. Accordig to thi model, a reeged cutomer ca be coviced i may cae by employig certai covicig mechaim to tay i the queue for completio of hi ervice. Thu, a reeged cutomer ca be retaied i the queuig ytem with ome probability (ay, q) ad it may leave the queue without receivig ervice with probability p (-q). Thi proce i referred to a cutomer retetio. We coider a igle erver, fiite capacity queuig ytem with cutomer retetio ad balkig i which the iterarrival ad ervice time follow egative-expoetial ditributio. The reegig time are aumed to be expoetially ditributed. A arrivig cutomer may ot joi the queue if there i at leat oe cutomer i the ytem, i.e. the cutomer may balk. The teady tate olutio of the model ha bee obtaied. Some performace meaure have bee computed. The eitivity aalyi of the model ha bee carried out. The effect of probability of retetio o the average ytem ize ha bee tudied. The umerical reult how that the average ytem ize icreae proportioately ad teadily a the probability of retetio icreae. Some particular cae of the model have bee derived ad dicued. Keyword Cutomer Retetio, Reegig, Balkig, Steady-State Solutio, Seitivity-Aalyi, Fiite Capacity. Itroductio I the curret ceario of populatio exploio ad globalizatio of iteratioal commerce ad trade, the queuig problem have gaied a lot of igificace i the deciio makig proce. Queuig theory ha revolutioized the idutry ad logitic ector apart from it immee applicatio i may other area like city traffic, air traffic, biociece, populatio tudie, health ector etc. Queuig model have bee built accordig to the prevailig demad or ituatio. The cocept of cutomer impatiece i queuig ha bee itroduced i 95 ad till people are exploitig thi cocept i variou applicatio[-]. A cutomer i aid to be impatiet if he ted to joi the queue oly whe a hort wait i expected ad ted to remai i the lie if hi wait ha bee ufficietly mall. Impatiece geerally take three form. The firt i balkig, the reluctace of a cutomer to joi a queue upo arrival, the ecod reegig, the reluctace to remai i lie after joiig ad waitig, ad the third jockeyig betwee lie whe each of a umber of * Correpodig author: rakeh_tat_kuk@yahoo.co.i (Rakeh Kumar) Publihed olie at Copyright Scietific & Academic Publihig. All Right Reerved parallel lie ha it ow queue[]. The otio of cutomer impatiece appeared i the queuig theory i the work of Haight[3] i 957. He coidered a model of balkig for M/M/ queue i which there i a greatet queue legth at which a arrival would ot balk. Thi legth wa a radom variable whoe ditributio wa ame for all cutomer. Haight[4] tudied a queue with reegig i which he tudied the problem like how to make ratioal deciio while waitig i the queue, the probable effect of thi deciio etc. Acker et al[5] tudied M / M / queuig ytem with balkig ad reegig ad performed it teady tate aalyi. Acker et al[6] alo obtaied reult for a pure balkig ytem (o reegig) by ettig the reegig parameter equal to zero. Queuig theory ha uccefully bee applied to variou cogetio (queuig) ituatio ivolvig reveue geeratio through ervicig cutomer. Aforemetioed queuig ytem deal with the cutomer lo due to impatiece (balkig or reegig) which reult ito a ubtatial reductio i the total reveue. Cutomer impatiece ha become the burig problem of private a well a govermet ector eterprie. They are cotatly workig toward cutomer retetio for better future propect. I thi paper, we coider a igle erver, fiite capacity queuig ytem with cutomer retetio ad balkig, i

2 Rakeh Kumar et al.: A M/M// Queueig Model with Retetio of Reeged Cutomer ad Balkig which the iter-arrival ad ervice time follow egativeexpoetial ditributio. Each cutomer upo arrivig i the queue will wait a certai legth of time for ervice to begi. If it ha ot begu by the, he will get impatiet ad leave the queue without gettig ervice. Thi time i a radom variable ad follow expoetial ditributio. A impatiet cutomer (due to reegig) ca be made to tay i ervice ytem for hi ervice by utilizig certai covicig mechaim. Such cutomer are termed a retaied cutomer. Whe a cutomer get impatiet (due to reegig), he may leave the queue with ome probability ay pp ad may remai i the queue with ome other probability q( p). A arrivig cutomer may ot joi the queue if there i at leat oe cutomer i the ytem, i.e. the cutomer may balk. The teady tate olutio of the model ha bee obtaied. Some queuig model have bee obtaied a particular cae of the model. The eitivity of the aalyi model ha bee carried out to how the impact of cutomer retetio probabilitie o the meaure of performace like expected ytem ize etc. A comparative tudy of the preet model with two other related model ha alo bee carried out. Ret of the model ha bee arraged a follow: ectio deal with the formulatio of tochatic queuig model, i ectio 3, the differetial-differece equatio of the model have bee made, i ectio 4, the teady-tate olutio of the model ha bee derived, ectio 5 deal with the eitivity aalyi of the model, ectio 6, deal with the particular cae of the model with comparative aalyi ad i ectio 7, the paper ha bee cocluded.. Stochatic Queuig Model The queuig model uder coideratio i baed o followig aumptio:. The arrival occur i a Poio tream oe by oe with a average arrival rate. The iter-arrival time are idepedetly, idetically ad expoetially ditributed with parameter.. There i oly oe erver ad ervice time are expoetially ditributed with parameter μ. 3. The queue diciplie i firt come, firt- erved (FCFS). 4. The capacity of the ytem i take a fiite (ay ). 5. Each cutomer upo arrivig i the queue will wait a certai legth of time (reegig time) for ervice to begi. If it ha ot begu by the, he will get impatiet ad may leave the queue without gettig ervice with probability p ad may remai i the queue for hi ervice with probability q( p). The reegig time follow expoetial ditributio with parameter. 6. The arrivig cutomer balk with probability /, where i the umber i ytem ad i the maximum umber allowed i the ytem. 3. Derivatio of Differetial-Differece Equatio Defie, P () t the probability that there are cutomer i the ytem, that i, i the queue ad oe i ervice. For, i a ifiiteimally mall iterval ( tt, + δt) P( t + δt) Pr{there are cutomer i the ytem at time ( t + δt) }. It ca happe i the followig mutually-excluive way: i. O arrival a cutomer either decide to joi the queue with probability (-/) or balk with probability / whe cutomer are ahead of him (,,..., ). The ytem might be i tate at time t ad durig the time iterval of legth δ t, o arrival, o departure ad o reegig occur. ii. The ytem might be i tate at time t, a arrival ad a departure both occur durig the time iterval of legth δ t. iii. The ytem might be i tate at time t ad a arrival take place durig the time iterval of legth δ t. iv. The ytem might be i tate + at time t, a departure take place ad o arrival occur durig the time iterval of legth δ t. v. The ytem might be i tate at time t ad durig the iterval of legth δ t, a reeged cutomer with probability q( p) doe ot abado the queue i.e. he tay i the ytem for hi ervice. vi. The ytem might be i tate + at time t ad durig the iterval of legth δ t a impatiet cutomer with probability p abado the queue. Hece, for P ( t + δt) P δt [ t][ ( ) t] µδ δ + P [( ) δt] + P + ( t)( µδt)[ ( ) δt] + P [( ) q] + P + ( t)[ pδt] + o( δt) Fidig the differece P( t + δt) P() t, dividig both ide by δ t ad takig limit δt lead to equatio (). o( δ t) approache to zero a rapidly a δt Similarly, other equatio ca be derived. The differetial-differece equatio of the model are: dp (t) P (t) + µ P (t) () dp ( ) p P + µ + () + [ µ + pp ] + + [( ) ] P ; dp () t [ ] P [ µ + ( ) pp ] ; (3)

3 America Joural of Operatioal Reearch, (): Steady-State Solutio dp () t I teady tate, limp () t P ad therefore t a t. Thu, the teady-tate equatio correpodig to equatio () - (3) are a follow: P + µ P (4) + µ + ( ) p P + [ µ + p] P + + [( ) ] P ; P [ µ + ( ) P ] ; (6) Solvig recurively equatio (4) (6), we get ( k ) P P ; (7) k µ + ( k ) Alo for, we get ( k ) P P ; (8) µ + ( k ) k Uig the ormalizatio coditio, P, we get P ( k ) + k µ + ( k ) Meaure of Performace () The Sytem Size k L P ( k ) P µ + ( k ) () The Queue Legth LL qq WW qq (3) The Waitig Time i the Sytem W (4) The Waitig Time i the queue Wq W µ 5. Seitivity Aalyi For 4 ad differet value of µ, & p, we have P 4 4 ( k ) + k 4 µ + ( k ) ( k ) P 4 µ + ( k ) k (5) (9) CASE-: Variatio of Sytem Size with the variatio i Average Arrival Rate Whe 4,...,.,...,.9, µ 3,.& q Sytem Size () V Average Arrival Rate () Figure. CASE-: Variatio of Sytem Size with the variatio i Average Service Rate Whe 4,, µ.,.,...,.9,.& q Sytem Size () V Average Service Rate (μ) Figure. CASE-3: Variatio of Sytem Size with the variatio i Average Reegig Rate Whe 4,, µ 3, q.6.,.,...,.9,.,.,..., Sytem Size () v Average Reegig Rate () Figure 3. Sytem Size () V Average Arrival Rate (). Sytem Size () V Average Service Rate (μ). Sytem Size () v Average Reegig Rate ().

4 4 Rakeh Kumar et al.: A M/M// Queueig Model with Retetio of Reeged Cutomer ad Balkig CASE-4: Variatio of Sytem Size with the variatio i Probability of Retetio Whe 4,, µ 3,.& q.,.,...,.9,.9,..9,..., Sytem ize()) V Probability of Retetio (q(-p)) Figure 4. From the figure ad, we ee that with the icreae i average arrival ad average ervice rate, there i expoetial icreae ad decreae i the expected ytem ize repectively. From fig.3 oe ca ee a regular decreae i average ytem ize with the icreae i average reegig rate. The mot importat i the effect of probability of retetio o the ytem ize. From fig.4 we ee that a we icreae the probability of retetio there i a teady ad proportioal icreae i the average ytem ize. 6. Particular Cae (i) Whe the probability of retetio, q The model reduce to M/M// queuig model with reegig ad balkig with ( k ) P P. () k µ + ( k ) Alo for we get ( k ) P P. () µ + ( k ) k Uig the ormalizatio coditio, P, we get P ( k ) + k µ + ( k ) Sytem Size P k ( k ) P µ + ( k ) Sytem ize()) V Probability of Retetio (q(-p)). () (ii) Whe the probability of reegig, p, implie : The model reduce to a imple M/M// queue with balkig with ( k ) ; k µ (3) P P Uig the ormalizatio coditio, P, we get Sytem Size 7. Cocluio P ( k ) ( + ) k µ L P (4) ( k ) k µ (5) P Thi paper dicued a Markovia igle erver fiite capacity queuig model with balkig ad poibilitie of retaiig reeged cutomer. The itroductio of thi cocept of cutomer retetio i queuig model ha a lot of igificace i the reveue geeratig queuig ytem. The teady-tate probabilitie of ytem ize have bee obtaied ad ome meaure of performace have alo bee computed. The effect of parameter like average arrival rate, average ervice rate etc. have bee tudied umerically ad it ha bee foud that with the icreae i probability of retetio, the expected ytem ize alo icreae proportioately ad teadily. Some particular cae of the model have bee dicued. REFERECES [] El-Paoumig,M.S., ad Imail, M.M.,9, O a trucated Erlag Queuig Sytem with Bulk Arrival, Balkig ad Reegig, Applied Mathematical Sciece,3(3),3-3 [] Al-Seedy, R.O., El-herbiy, A.A., El-Shehawy, S.A, ad Ammar, S.I.,9, Traiet Solutio of the M/M/c queue with balkig ad reegig, Computer ad Mathematic with Applicatio, 57(8), 8-85 [3] Cochra, J.K., ad Broyle, J.R.,, Developig oliear queuig regreio to icreae emergecy departmet patiet afety: Approximatig reegig with balkig, Computer ad Idutrial Igeeerig,59(3), [4] Dow, D.G., Koole,G., ad Lewi, M.E., Dyamic cotrol of a igle erver ytem with abadomet, Queuig Sytem, 67(), 63-9 [5] Choudhury, A., ad Medhi, P.,, Balkig ad reegig i multierver Markovia queuig ytem, Iteratioal Joural of Mathematic i Operatioal Reearch, 3(4), [6] Boxma, O., Perry D., ad Stadje,V.,, The M/G/+G queue reviited, Queuig Sytem,67(3),7-

5 America Joural of Operatioal Reearch, (): -5 5 [7] Kapoditria, S.,, The M/M/ queue with ychroized abadomet, Queuig Sytem,68,79-9 [8] Liao, P.,, A queuig model with balkig idex ad reegig rate, Iteratioal joural of Service ad operatio maagemet,(),- [9] Kruk, L., Lehoezky, J., Ramaa, K., ad Shreve, S.,, Heavy traffic aalyi for EDF queue with reegig, The Aal of Applied Probability,(), [] Tarabia, A. M. K.,,Traiet ad teady tate aalyi of a M/M/ queue with balkig,catatrophe, erver failure ad repair, Joural of Idutrial ad Maagemet Optimizatio,7(4),8-83 [] Khodabi, I. G. ad Fathi, M.,, uig queuig approach for locatig the order peetratio poit i a two-echelo upply chai with cutomer lo, Iteratioal Joural of Buie ad Maagemet, 6(),58-68 [] Gro, D. ad Harri C. M.,974,Fudametal of Queuig theory, Joh Wiley ad So,34-4 [3] Haight, F. A., 957, Queuig with balkig, I, Biometrika, 44, [4] Haight, F. A.,959, Queueig with Reegig, Metrika,86-97 [5] Acker Jr., C. J. ad Gafaria, A. V., 963, Some Queuig Problem with Balkig ad Reegig. I, Operatio Reearch, (),88- [6] Acker Jr., C. J. ad Gafaria, A. V., Some Queuig Problem with Balkig ad Reegig. II, Operatio Reearch,(6),

4.6 M/M/s/s Loss Queueing Model

4.6 M/M/s/s Loss Queueing Model 4.6 M/M// Lo Queueig Model Characteritic 1. Iterarrival time i expoetial with rate Arrival proce i Poio Proce with rate 2. Iterarrival time i expoetial with rate µ Number of ervice i Poio Proce with rate