Time Varying Multiserver Queues. W. A. Massey. Murray Hill, NJ Abstract

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1 Waiing Time Aympoic for Time Varying Mulierver ueue wih Abonmen Rerial A. Melbaum Technion Iniue Haifa, 3 ISRAEL avim@x.echnion.ac.il M. I. Reiman Bell Lab, Lucen Technologie Murray Hill, NJ 7974 U.S.A. mary@reearch.bell-lab.com W. A. Maey Bell Lab, Lucen Technologie Murray Hill, NJ 7974 U.S.A. will@reearch.bell-lab.com A. L. Solyar Bell Lab, Lucen Technologie Murray Hill, NJ 7974 U.S.A. olyar@reearch.bell-lab.com Abrac We conider a nonaionary Markov mulierver queueing model where waiing cuomer may abon ubequenly rery. In hi paper we derive uid diuion approximaion for he aociaed waiing ime proce. The uid diuion approximaion for he correponding queue lengh proce were obained in [4] (ee alo [5]). Inroducion The model we conider in hi paper i a muli-erver queue wih ime-varying parameer, in which cuomer are impaien hence abon afer (ubjecively) exceive wai. Moreover, obaining ervice i imporan enough for ome cuomer ha hey reurn eek ervice (rery) afer a \ime-ou". Formally, our model i depiced in Figure : here i a ingle \ervice" node wih n,, erver. New cuomer arrive o he ervice node following a Poion proce of rae. Cuomer arriving o nd an idle erver are aken ino ervice ha ha rae. Cuomer ha nd all erver buy join a queue, from which hey are erved in a FCFS manner. Each cuomer waiing in he queue abon a rae. An aboning cuomer leave he yem wih probabiliy or join a rerial pool wih probabiliy,. Each cuomer in he rerial pool leave o ener he ervice node a rae. Upon enry o he ervice node, hee cuomer are reaed he ame a new cuomer. The behavior of he yem i decribed by he wo-dimenional, coninuou ime Markov chain () = (); () where () equal he number of cuomer reiding in he ervice node (waiing or being erved) () equal he number of cuomer in he rerial pool. Our work i moivaed by he need o develop analyical ool ha uppor performance analyi of large elecommunicaion yem, uch a elephone call cener, where

2 µ () λ ()... µ ( () n ) n β ψ ( () n ) + β ( ψ ) ( () n ) + ()... 8 Figure : The abonmen queue wih rerial. abonmen rerial arie naurally. Call cener are conanly ubjec o imevarying condiion, waiing cuomer in phone queue are unable o oberve he ae of he yem. I follow ha ime-dependen modeling (a oppoed o alo aedependen) i naural for call cener. Finally, we poin ou ha he analyi of waiing ime i (ypically) analyically more challenging han ha of he queue lengh, in many applicaion (like call cener) i probably more imporan. For more dicuion relaed reference on hee iue, ee [5]. The remainder of hi paper i organized a follow. The aympoic regime we conider i inroduced in Secion. In Secion 3 we provide uid diuion limi for he virual waiing ime a a xed ime. Proce level uid diuion limi for he virual waiing ime are preened in Secion 4. Aympoic Regime A menioned above we are inereed in he behavior of a yem wih large number of erver large inpu rae. Thu, we conider he aympoic regime where we cale up he number of erver in repone o a imilar caling up of he arrival rae by cuomer. More preciely, he aympoic regime i a follow. Aume ha ; ; ; ; ;n are xed funcion of ime. We conider a equence of yem indexed by caling parameer = ; ;:::, k!a k!.(toavoid cumberome noaion, in wha follow, we index a yem by, when we wrie!,we mean ha goe o inniy by aking value from he equence ; ; :::.) In a yem wih index, he arrival rae (i.e., he ineniy of he Poion arrival proce) i he number of

3 erver i n. (Acually, he laer hould be, for example, he ineger par of n, bu again, o avoid rivial complicaion implify noaion, we aume i' ju n.) We alo make he following addiional Aumpion. The funcion n i coninuouly diereniable in [; ). Sample pah of he family of queue lengh procee () = (); (), indexed by he caling parameer, are deermined by he following equaion: () = () a + c () d, b (), n + (, )d b + d, (), n + c d, () ^ n () = () + b (), n + (, )d, c () d ; (.) where a ; b ; c ; b ;c ; are independen ard (rae ) Poion procee. In hi paper we ue he noaion x ^ y = min(x; y) x + = max(x; ) for all real x y. Throughou hi paper we aume ha he following iniial condiion hold: where () () () () are xed vecor, ()! () () ; (.3),= [ (), () ()]! () () ; (.4) () () > : (.5) In he re of he paper we alo ue he following noaion. Le E be a complee eparable meric pace, a be a real number. Then we denoe by D(E;a) he Skorohod pace of E-valued funcion dened in he inerval [a; ) which are righ coninuou have lef limi. The pace D(E;a) i endowed wih Skorohod J -meric he correponding opology. 3 Waiing Time in Node : Marginal Diribuion a a Given Time. Suppoe ha we are inereed in he waiing ime of a \virual" cuomer arriving a aion a a xed ime. Since we have a yem wih abonmen, a convenien way o approach hi problem i o conider he yem ha i obained from he original one by he following modicaion. Suppoe, ha afer ime, here are no new exogenou arrival ino he yem, any cuomer deparing any aion i leave he yem. In oher word, aring ime, each aion i ha no new arrival, i ju erve he cuomer which were a he aion a ime. Theorem 5. in [4] ill applie o he modied yem; he only dierence i ha he erm in he equaion, correponding o he arrival afer ime, hould be \zeroed ou". Namely, he following reul follow direcly from Theorem 5. ( i proof) in [4]. Denoe he arrival deparure procee for aion by A = f A () j g = f () j g (.) d

4 repecively. Le, by convenion, he arrival proce include he cuomer in node a ime, o A () = (), () =, A (), () = ();. Then we obain he following uid limi reul. Theorem 3. Wih probabiliy, he following convergence hold uniformly on compac e (u.o.c.) of : ( ;A ; )! ( () ;A () ; () ) (3.) where =( ; ); () =( () ; () ), he uid limi () aie he following equaion () () = () () + h + () () i fg, () () ^ n, () (), n + d (3.) () () = () ^ () + (, ) () + (), n d, () ()d : (3.3) Moreover, A () () are equal o () () = A () () = () () + () ^ () ^ n + () h + () () i d (3.4) (), n + d ; (3.5) where () iaconinuouly diereniable non-decreaing funcion in [; ). We alo obain he following diuion limi. Theorem 3. The following weak convergence hold (in he pace being he direc produc of correponding Skorohod pace D(R; )) : p (, () ; A, A () ;, () ) d! ( () ;A () ; () ); (3.6) where () =( () ; () ) i he unique coninuou oluion o he ochaic dierenial equaion () () = () () + ^ +,B b,b c () () = () () + +B c h () (), () () + () ^ + B a ^ i () d (3.7) () ()d, B c () () d d () (), n + b (, () )d, B (), n + d () () ^ n d ^ () ^ () (, )d, () () d + B b () ^ ()d (3.8) () (), n + (, )d ;

5 wih () () = () () + () f where A () () are dened a A () () = () ()+ ()= () h Clearly, +B b ^ () ()d,b c, () ()ng ^ (), f () () (), () () + i () () d + B c b + B () (), n + (, )d ()>ng; (3.9) a () () ^ d +B d (3.) () () ^ n d (3.) () (), n + d : () () =A () (), () () : (3.) Now, le u dene he \poenial ervice iniiaion" proce D for node by D () = ()+n ; : Noe ha if () <n, hen A () <D (); o he poenial ervice can be \ahead" of arrival. Obviouly, we have he (probabiliy, u.o.c.) convergence: D ()! D () (); ; where D () () = () ()+n ;. Since n i coninuouly diereniable by aumpion we know ha () () iconinuouly diereniable, D () () i alo coninuouly differeniable we denoe i derivaive by d () (). Now we will make an imporan (bu no very rericive in majoriy of applicaion) addiional aumpion. Aumpion 3.. The funcion D () (of ) iconinuouly diereniable wih ricly poiive derivaive, lim! D() () >A () () : (3.3) (Noe, ha according o our deniion, boh A () A () () are conan inhe inerval [;).) Alo, i will be convenien o adop a convenion ha all he procee we conider are dened in he inerval [,T;), wih T = n =d () () : We make hi exenion by auming ha nohing i happening in he inerval [,T;) (no arrival or deparure) excep he number of erver i increaing linearly from o n (for he uncaled proce wih index ). We hen can rewrie (3.) (3.6) a follow (wih all he funcion being now dened for,t ): ( ;A ;D )! ( () ;A () ;D () ) (3.4)

6 A () ()+A () () A () () A () () A () W () () D () (+W () ()) () () n A () () D () (+W () ()) D () D () D () +D () d () (+W () ()) + W () () +W () () Figure : The diuion erm for he aainmen waiing ime where p (, () ; A, A () ; D, D () ) d! ( () ;A () ;D () ) ; (3.5) D () = () : (3.6) Noe ha procee A () ;D () ;A () ;D () are coninuou D () (,T )=D () (,T )=. Our convenion ogeher wih he Aumpion 3. make he following procee well dened nie wih probabiliy for all ucienly large. Le u dene, for all,t, he r aainmen procee he aainmen waiing ime procee S () = inff,t : D () >A ()g S () () = inff,t : D () () >A () ()g; (3.7) W () =S (), W () () =S () (), : (3.8) Denoe by ^W () he virual waiing ime a, i.e. he ime a \e" cuomer (in he original non-modied yem) arriving in node a ime would have o wai unil i ervice ar, auming hi cuomer doe no abon while waiing. Then he relaion beween he virual waiing ime ^W () he aainmenwaiing ime W () i imply ^W () =W () + : (3.9)

7 Indeed, noe ha W () ( W () ()) may be negaive. All hi mean i ha () < n, herefore in hi cae ^W () =. IfW () i non-negaive, hen i value i exacly equal o he virual waiing ime. I follow direcly from Theorem Corollary in [7] ha (3.4), (3.5), Aumpion 3., imply he following convergence. Wih probabiliy, u.o.c., In diribuion, ( ; A ; D ;W )! ( () ;A () ;D () ;W () ) : (3.) p (, () ; A, A () ; D, D () ;W, W () ) d! ( () ;A () ;D () ;W () ) ; (3.) where W () () = A() (), D () (S () ()) d () (S () ()) Since he procee A () ;D () ; () ;W () are coninuou wih probabiliy,we auomaically obain he weak convergence of nie dimenional diribuion. In paricular, conider he non-rivial cae S () () (which i equivaleno () () n ). We obain W ()! W () () p (W (), W () ()) d! W () () = () (S () ()) d () (S () ()) : Solving equaion (3.) for () () in he inerval [;), we obain () () = () () exp We can nd S () () from, d + exp, r dr S () () = minf j () () =n g : : (, )n d ; : Solving a ochaic dierenial equaion for () () in he inerval [;S () ()], we obain (cf. [] () (S () ()) d = () () exp, where f S () ( ) d! + S() ( ) exp =(() (), n ) + n ; B i a ard Brownian moion proce. In paricular, E[ () (S () ())] = E[ () ()] exp, Var[ () (S () ())] = Var[ () ()] exp, S () ( )! r dr S () ( ), + S() ( ) r dr! d exp, S() f db(, ) ; ( )! r dr f d :

8 Noe ha in cae () () =n,we obain S () () =; W () () =; d () () = n + n ;, herefore, p W ()! d W () () = () () n + n : Recalling (3.9), we obain he following diuion limi for he virual waiing ime in hi cae p ^W ()! d () () + n + n ; if () () =n ; which i wha we inuiively expeced. We checked he accuracy of he uid approximaion for he virual waiing ime via imulaion. The yem we conidered ha all parameer conan excep for. In paricular we conidered n = 5, =, =:, =:5, =:5, wih =+, ;. The reul are hown in Figure 3. The graph on op compare he uid imulaion reul for he queue lengh, he graph on he boom compare he uid imulaion reul for he virual waiing ime. (The imulaion reul depiced are an average of 5 independen replicaion. More deail on he imulaion mehod are conained in [5].) 4 Waiing Time in Node : A Proce In he previou ecion we derived uid diuion approximaion of he marginal diribuion of he aainmen waiing ime, which uniquely deermine hoe for he virual waiing ime, in node a a given ime. A naural conjecure i ha one can obain imilar aympoic for he aainmen waiing ime a a rom proce dened for [; ). In hi ecion we preen reul howing ha he above conjecure i indeed rue. We need more deniion. Fir, in hi ecion, unle oherwie explicily aed, we view all he procee a rom procee of wo ime variable, [,T;) [; ). (In he previou ecion wa a xed parameer.) More preciely, we view hem a rom elemen X =((X(; ); [,T;)); ) (X can be i or A or (j) i, ec.) aking value in he pace D(D(R;,T ); ). Noe ha for each xed all procee of inere are well dened in he previou ecion, he convergence (3.) (3.) do hold for any xed. Aumpion 4. Aumpion 3. hold for any. A generalizaion of he argumen ued in he proof in [4] (roughly, making all eimae in he convergence proof \uniform on "), a generalizaion of he reul in [7], lead o he following reul which are exenion of (3.) (3.). The deail are conained in [6]. Fir, we ae our funcional rong law of large number reul. Theorem 4. Wih probabiliy, uniformly on compac e of (; ), ( ; A ; D ;S ;W )! ( () ;A () ;D () ;S () ;W () ) ; (4.) where all funcion (), A (), D (), S (), W (),areconinuou joinly on, for each xed hey (a funcion of ) aify he ODE (3.), (3.3), equaion (3.4), (3.5), (3.7), (3.8). Moreover, d () (; ) (@=@)D () (; ) i ricly poiive.

9 3 lambda = +, n = 5, mu =., mu =., bea =.5, P(rerial) =.5 5 q fluid q im q fluid q im queue lengh ime 3.5 lambda = +, n = 5, mu =., mu =., bea =.5, P(rerial) =.5 3 vw fluid vw im.5 virual waiing ime ime Figure 3: The uid approximaion for he queue lengh virual waiing ime

10 Now we ae our funcional cenral limi heorem. Theorem 4. The following weak convergence hold: p (, () ; A, A () ; D, D () ; (W (;), W () (;); )) d! ( () ;A () ;D () ; (W () (;); )) ; (4.) where () i a joinly coninuou on rom proce, which (a a funcion of, wih xed) i he unique oluion o he ochaic dierenial equaion (3.7) (3.8); A () D () (a funcion of ) aify (3.), (3.), (3.6), are joinly coninuou on ; W () (;)= A() (;), D () (S () (;);) d () (S () (;);) i coninuou on. 5 Acknowledgemen The auhor hank Brian Rider of Couran Iniue for hi work on he numerical imulaion, compuaion, plo of he graph hown in Figure 3. Reference [] S. N. Ehier T. G. Kurz. Markov Proce: Characerizaion Convergence. John Wiley Son, New York, 986. [] I. Karaza S. E. Shreve. Brownian Moion Sochaic Calculu (Second Ediion). Springer-Verlag, New York, 99. [3] T. G. Kurz. Srong approximaion heorem for deniy dependen Markov chain. Sochaic Procee Their Applicaion, 6:3{4, 978. [4] A. Melbaum, W. A. Maey, M. I. Reiman. Srong Approximaion for Markovian Service Nework. ueueing Syem, (998). [5] A. Melbaum, W. A. Maey, M. I. Reiman, B. Rider. Time Varying Mulierver ueue wih Abonmen Rerial. ITC-6, Edinburgh, Scol, (999). [6] A. Melbaum, W. A. Maey, M. I. Reiman, A. L. Solyar. Waiing Time Aympoic for Mulierver, Nonaionary Jackon Nework wih Abonmen. In preparaion. [7] A. Puhalkii. On he Invariance Principle for he Fir Paage Time. Mahemaic of Operaioin Reearch, Vol. 9, (994), pp