On the relationships among queue lengths at arrival, departure, and random epochs in the discrete-time queue with D-BMAP arrivals

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1 On the relatonshps among queue lengths at arrval departure and random epochs n the dscrete-tme queue wth D-BMAP arrvals Nam K. Km Seo H. Chang Kung C. Chae * Department of Industral Engneerng Korea Advanced Insttute of Scence and Technolog Daejon Korea Abstract We consder fnte- and nfnte-capact queues wth dscrete-tme batch Marovan arrval processes (D-BMAP) under the assumpton of the Late Arrval Sstem wth Delaed Access as well as the Earl Arrval Sstem. Usng smple arguments such as the balance equaton rate n = rate out we derve relatonshps among the statonar queue lengths at arrval at departure and at random epochs. Such relatonshps hold for a broad class of dscrete-tme queues wth D-BMAP arrvals. Kewords: Dscrete-tme queues; Marovan arrval process; Queue length * Correspondng author. Tel: ; fa: E-mal address: cchae@ast.ac.r Acnowledgements: The authors are grateful to Dr. Mohan L. Chaudhr and the anonmous revewer for ther constructve comments that greatl mproved the presentaton of ths paper.

2 . Introducton and notaton In recent ears there has been a growng nterest n the analss of dscrete-tme queues due to ther applcatons n slotted dgtal communcaton sstems and other related areas. One of the reasons for ths fact s that dscrete-tme queueng models ft better the dscrete nature of computer and communcaton sstems than the contnuoustme counterparts and therefore the can gve more accurate performance measures of these sstems. In ths paper we consder the statonar queue lengths of fnte- and nfnte-capact queues wth dscrete-tme batch Marovan arrval processes (D-BMAP). Usng smple arguments such as rate n = rate out we derve relatonshps among the statonar queue lengths at arrval at departure and at random epochs. Relatonshps of ths nd are useful n the sense that once one obtans a soluton for certan epochs e.g. departure epochs t s mmedate to get solutons for the others through these relatonshps. Contnuous-tme counterparts of these relatonshps are avalable n Tane and Taahash [3] and Km et al. [8]. Remar. The D-BMAP s a versatle process that contans a large class of dscretetme pont processes such as Bernoull arrval process Marov modulated Bernoull process and batch Bernoull process wth correlated batch arrvals. (see e.g. Blonda and Casals [2]). Furthermore we do not assume an partcular servce mechansm e.g. the servce dscplne the number of servers the batch servce generalzed vacatons etc. Thus the relatonshps presented n ths paper hold for a broad class of dscrete-tme queues wth D-BMAP arrvals. In dscrete-tme queueng models the tme as s dvded nto fed-length ntervals called slots and customer arrvals and departures are assumed to be tang place at slot boundares. Dfferent assumptons can be made on the order of an arrval and a departure smultaneousl tang place at a slot boundar: ether an arrval ma have precedence over a departure or vce versa. The former case s referred to as the Late Arrval Sstem (LAS or arrvals frst polc) and the latter as the Earl Arrval Sstem (EAS or departures frst polc). Untl turnng our attenton to the EAS n Secton 4 we

3 assume the LAS wth Delaed Access (LAS-DA) where delaed access means that arrvng customers who fnd the server(s) avalable do not mmedatel get served but are delaed untl the begnnng of the net slot. For more detals on these models see Bruneel and Km [4 p.] Hunter [7 p.93] and Taag [2 p.4]. In a queue wth D-BMAP arrvals groups of customers of sze arrve at the queue accordng to a D-BMAP wth representaton { 0} D where D s an m m matr. Note that m denotes the number of phases n the underlng Marov chan (UMC) that governs the arrval process. Suppose that the UMC s n some phase n the mddle of a slot. Then under the LAS-DA wth respectve probabltes (D ) j and ( D 0) j there s a phase transton to j wth a batch arrval of sze and wthout an arrval just pror to the end of the slot. Just after the slot boundar there ma be a departure of customer(s) f an. Note that nothng could happen somewhere n the mddle of a slot (for more detals on the D-BMAP see Blonda [] [2] and Herrmann [6]). Let λ and λ g respectvel denote the average numbers of customer arrvals and batch arrvals per slot. The are then gven b = λ = π D e = λ = π D e g where π s the statonar probablt vector of the UMC wth the transton probablt matr D = D = 0 and e s a column vector of s. Let N be the upper lmt on the number of customers n the sstem. We assume the Partal Rejecton Polc (PRP) n whch arrvng customers n a batch fll empt postons n the sstem and once all avalable postons are flled the remanng customers n the batch are rejected (see e.g. Taag [ p. 42]). Now we consder a dscrete-tme bvarate process { L S ; } where L denotes the number of customers n the sstem and S denotes the phase of the UMC both n the mddle of the th slot. We defne the followng probabltes n numberaverage sense for 0 N and j m :

4 : statonar probablt that the process s n state ( j) n the mddle of a slot j = ( m ) (We nterpret as the long-run fracton of slots durng whch the state s ( j).) j j g = : statonar probablt that a batch (or group) fnds customers n the sstem and that the UMC s n phase j just after ts arrval g ( m ) (We nterpret as the long-run fracton of batches that fnd customers n the j sstem and mae transtons of the UMC nto j.) j = : statonar probablt that an ndvdual customer fnds customers n the sstem (ncludng accepted customers who precede her n her own batch see Remar 2 below) and that the UMC s n phase j just after the arrval of her batch ( m ) (We nterpret j as the long-run fracton of ndvdual customers who fnd customers n the sstem wth the phase of the UMC beng j just after the arrvals of ther batches.) j : statonar probablt that an ndvdual customer leaves behnd customers n the sstem (ncludng served customers f an who follow her to depart at the same tme see Remar 2 below) and that the UMC s n phase j just after her departure = ( m ) (We nterpret as the long-run fracton of ndvdual customers who leave behnd j customers n the sstem wth the phase of the UMC beng j just after her departure.)

5 Remar 2. Note that we are adoptng a well nown conventon (see e.g. Wolff [4 p.388]) that customers of an arrvng/departng batch enter/leave the sstem not smultaneousl but one at a tme nstantaneousl. That s we suppose that customers of an arrvng batch form a lne to enter the sstem one after another. Lewse when we consder bul departures n batch-servce or mult-server queues we suppose that served customers departng at the same epoch form a lne to leave the sstem one after another. Partcularl ndvdual customers n such an arrvng lne who fnd N customers n the sstem are assumed not to enter the sstem but to mmedatel depart from the sstem leavng those N customers behnd. Note that we tae such departures b rejected customers nto account as well as those b accepted customers so that the average number of customer departures per slot s stll λ. 2. A relatonshp between the queue lengths at random and at departure epochs In ths secton we derve a relatonshp between the statonar queue lengths at random epochs n the mddle of slots and just after departure epochs b usng the balance equaton rate n = rate out. Hereafter the average numbers of transtons nto and out of some state ( j) per slot are called the transton rates nto and out of ( j). Especall for such queues as bul-arrval bul-servce and mult-server queues n whch customers ma arrve and/or depart n group we adopt the followng conventon to mae the process { L S ; } L : sp-free wth respect to Sp-free conventon: When a batch arrval of sze l > maes a transton from ( j) to ( + l ) we suppose that ths transton nstantaneousl goes through a sequence of n-between states ( + ) ( + 2 ) ( + l ). That s a gant transton from ( j) to ( + l ) s supposed to be made up of l nstantaneous onestep transtons: ( j) to ( + ) ( + ) to ( + 2 ) ( + l ) to ( + l ). Smlarl when a batch departure of sze l > maes a transton from ( j) to ( l j) we suppose that ths gant transton nstantaneousl goes through a

6 sequence of n-between states ( j) ( 2 j) ( l + j). We need ths conventon n order to represent transton rates nto and out of some state ( j) n terms of such quanttes as j (and j ) whch are defned n an ndvdual-customer-average sense (note that ther defntons correspond not to gant transtons but to one-step transtons). Now we equate the transton rates nto and out of state ( j) assumng that the est. As a result we have Theorem. For a fnte-capact queue wth D-BMAP arrvals under the assumptons of LAS-DA and PRP and are related b () D = 0 = λ λ 0 N () N () D λ. (2) = 0 n= N n N = N Before presentng a proof of ths theorem we gve an essental argument for calculatng the rate of the transtons caused b customer departures. Note that λ s the average number of ndvdual customer departures per slot (ncludng departures b rejected customers) and that N s the fracton of ndvdual customers j who leave behnd the sstem wth state ( j). That s λ s the average number of ndvdual customers per slot who depart to mae transtons from ( j) to ( j) note that such a transton ma be a part of a gant transton. Thus λ j s consdered to be the transton rate out of ( j) as well as the transton rate nto ( j) caused b ndvdual customer departures. Note also that there are λ rejected customers per slot who leave behnd the sstem wth state ( N j ). Ther departures do not mae an transtons at all because rejected customers can not mae an effects on the state of the process ecept the frst customers of rejected batches who can mae phase-transtons of the UMC at ther arrvals. j N j

7 Proof. For the process { L S ; } we note that transtons nto and out of ( j) durng a slot are caused ether b a phase transton of the UMC (wth or wthout an arrval) or b a departure. () The case 0 N We frst consder the transton rate out of ( j). The out-rate caused b phase transtons of the UMC s gven b j( ( D 0) jj) + X where X s the rate of gant transtons nstantaneousl gong through ( j). (That s X s the transton rate from {( n l) n> 0 l m} to {( n j) n 0} + > caused b batch arrvals. Under spfree conventon ths rate of nstantaneous transtons should be consdered as a part of the transton rates nto and out of ( j). However we do not have to elaborate more on X because t cancels out n the balance equaton.) In addton the rate out of ( j) caused b departures s gven b λ j as dscussed earler. Thus we have for 0 N and j m transton rate out of ( j) where 0. j = { } ( ( ) ) = D + X +λ (3) j 0 jj j Net we consder the transton rate nto ( j). Note that the rate nto ( j) caused b phase transtons of the UMC s gven b the total transton rate from ( l) to ( j) for all ( l) ( j) plus X the gant transton rate nstantaneousl gong through ( j). In addton the rate nto ( j) caused b departures s gven b λ j. Thus we have for 0 N and j m m transton rate nto ( j) = l ( D ) lj j ( D 0 ) jj + X + λ j. (4) = 0 l= Fnall () follows from equatng (3) and (4). () The case = N In a smlar manner we frst consder the transton rate out of ( N j). The out-rate caused b phase transtons of the UMC s gven b the total transton rate from ( N j) to ( N l) for all l j. Note that n ths case there are no gant transtons havng ( N j) as an n-between state. In addton there are λ N j customer-departures per

8 slot that mae transtons out of ( N j). Thus we have transton rate out of ( N j) = N j ( D ) jj + λn j. (5) = 0 Smlarl the rate nto ( N j) caused b phase transtons of the UMC s gven b the total transton rate from ( l) to ( N j) for all ( l) ( N j). And customer departures cannot mae a transton nto ( N j) as dscussed earler. Thus we have N m transton rate nto ( N j) = l ( Dn) lj N j ( D ) jj. (6) = 0 n= N l= = 0 Fnall (2) follows from equatng (5) and (6). Note that λ 0 N of () and (2) can be replaced b λ * * where * λ = λ( e ) s the effectve arrval (or departure) rate and * = ( e ) N N represents the probablt of the queue length left behnd b an accepted customer. When the arrval process s a Bernoull process wth parameter λ (n ths case D 0 = λ D = λ and D = 0 2 ) () and (2) smplf to = 0 N. Now multplng both sdes of () b z and summng up for all 0 we have Corollar. For a stable nfnte-capact queue wth D-BMAP arrvals under the assumpton of LAS-DA Y ( and X ( are related b Y( ( D( I) = λ ( z ) X( (7) where Y ( z ) = z X ( z ) = z and D ( z ) = D z. = 0 = 0 = 0 We remar that Theorem and Corollar are dscrete-tme analogues of the contnuous-tme BMAP results. For eample (7) s the dscrete-tme counterpart of the contnuous-tme result Y( D( = λ ( z ) X( [8 3].

9 3. Relatonshps between the queue lengths at random and at arrval epochs In ths secton we present relatonshps between the statonar queue lengths at random and at arrval epochs. We vew an arrval n two respects: a batch arrval and an ndvdual-customer-arrval. Frst we present a relatonshp between the statonar queue length at random epochs n the mddle of slots and that found b batch arrvals as follows: Theorem 2. For a fnte-capact queue wth D-BMAP arrvals under the assumptons of LAS-DA and PRP and are related b λ = ( D D0 ) 0 N. (8) g Proof. We note that λ g s the average number of batch arrvals per slot and that j s the fracton of batches that fnd customers n the sstem and mae transtons of the UMC nto j. That s λ s the average number of batch arrvals of such nd g j per slot. Ths number obvousl equals the transton rate out of {( l) l m} caused b batch arrvals that mae transtons of the UMC nto j. Thus we have for 0 N and j m λ m g j l( ) lj = l= = D from whch the desred result follows. We note that Theorem 2 s stll vald for the Total Rejecton Polc (TRP) n whch all customers n an arrvng batch are rejected f the number of avalable postons s less than the batch sze (see e.g. [ p. 42]). Now multplng both sdes of (8) b z and summng up for all 0 we have Corollar 2. For a stable nfnte-capact queue wth D-BMAP arrvals under the assumpton of LAS-DA Y ( and Y ( are related b

10 Y D D ( = Y( λ g 0 where Y ( = z. = 0 Now we present a relatonshp between the statonar queue length at random epochs n the mddle of slots and that found b ndvdual customers (ncludng accepted customers who precede them n ther own batches). Theorem 3. For a fnte-capact queue wth D-BMAP arrvals under the assumptons of LAS-DA and PRP and are related b () () = = 0 n= + λ D 0 N (9) N N = 0 n= N+ n λ = ( n N) D. (0) n Proof. () The case 0 N We note that λ s the average number of ndvdual customer arrvals per slot (ncludng rejected customers) and that s the fracton of ndvdual customers who j fnd customers n the sstem wth the phase of the UMC beng j just after the arrvals of ther batches. That s λ s the average number of ndvdual customer j arrvals of such nd per slot. Ths number obvousl equals the average number of batch arrvals per slot that mae transtons from { l) l m} {( j) n + } n. Thus we have for 0 N and j m m j l( n) lj = 0 n= + l= λ = D ( to from whch (9) follows. () The case = N Under the PRP all rejected customers fnd N customers n the sstem when the arrve (see Remar 2). Thus we have n a smlar manner

11 N m N j= ( ) l( n) lj = 0 n= N+ l= D λ n N from whch (0) follows. Note that N e ( = Ne) s the probablt of an ndvdual customer beng rejected. Also note that when the arrval process s a Bernoull process wth parameter λ (9) and (0) smplf to = 0 N whch s a well nown propert called BASTA (Bernoull arrvals see tme averages Boma and Groenendj [3]). Now multplng both sdes of (9) b z and summng up for all 0 we have Corollar 3. For a stable nfnte-capact queue wth D-BMAP arrvals under the assumpton of LAS-DA Y ( and Y ( are related b D D( Y ( = Y( λ( where Y ( z ) = z. = 0 We remar that the relatonshps n Theorems 2 and 3 and Corollares 2 and 3 are the same as the contnuous-tme BMAP counterparts [8]. 4. Relatonshps n the Earl Arrval Sstem (EAS) In ths secton we consder the EAS (or departures frst polc) where customers arrve earl durng a slot. Suppose that the UMC s n some phase at an eact slot boundar. Then under the EAS wth respectve probabltes (D ) and j ( D 0) j there s a phase transton to j wth a batch arrval of sze and wthout an arrval just after the slot boundar. Just pror to the end of the slot there ma be a departure of customer(s) f an. Note agan that nothng could happen somewhere n the mddle of a slot. (For more detals on the EAS see [7 p.93] and [2 p.8].)

12 In addton to the probabltes defned n Secton we defne for the EAS the followng probabltes n number-average sense for 0 N and j m : w : statonar probablt that the process s n state ( j) at an eact slot boundar j w = ( w w m ) (We nterpret w as the long-run fracton of slot boundares at whch the process s n j state ( j).) Note that the values of and w under the assumpton of the EAS are n general dfferent from those under the LAS-DA (see Remar 3 below). That s the depend on whch assumpton the are under. Ther relatonshps however are smlar to each other as dscussed below. We note frst that transton epochs of the UMC just le w under the EAS corresponds to the queue length just before under the LAS-DA. Then we replace LAS- DA j and Y ( n Sectons 2 and 3 wth EAS w j w and W ( z ) = w z =0 respectvel. Wth ths replacement t s not dffcult to see that all the theorems and corollares stated for the LAS-DA stll hold for the EAS as well. Fnall the followng relatonshp between obvous (we omt the proof): w and both under the EAS s Theorem 4. For a fnte-capact queue wth D-BMAP arrvals under the assumptons of EAS and PRP w and are related b () = w D 0 N = 0 () N = w D. N = 0 n= N n In case of a stable nfnte-capact queue Y ( and W ( are related b () Y ( = W( D(.

13 Remar 3. For a smple model such as an nfnte-capact queue wthout vacatons under the EAS s the same as that under the LAS-DA. Ths s because the queue length n the mddle of a slot s not affected b the order of the arrval and the departure at the precedng slot boundar [2 p.38]. That s sample paths durng a slot under the two assumptons are eactl the same [7 p.94]. Ths s not the case however when t comes to fnte capact queues or queues wth vacatons. In such queues customers who can be accepted under the EAS ma not be accepted under the LAS or the server elgble for a vacaton under the EAS ma not be under the LAS. That s sample paths ma evolve dfferentl accordng to the assumpton of the order of the arrval and the departure smultaneousl tang place at a slot boundar. 5. Concludng Remars For fnte- and nfnte-capact queues wth D-BMAP arrvals we derved relatonshps among and 0 N under the LAS-DA (or EAS) and PRP. Because we dd not assume an partcular servce mechansm these relatonshps hold for a broad class of dscrete-tme queues wth D-BMAP arrvals. Suppose one has the soluton for 0 N of a D-BMAP/G ab /c/gv queue wth the threshold value of actvatng the server a the mamum sze of a batch for servce b the number of servers c and generalzed vacatons (GV). (Usuall of a smple model can be obtaned b consderng the embedded Marov chan at departures (see e.g. [ 6])). Then through the relatonshps t s mmedate to have and Wthout these relatonshps such as () for eample t ma requre length calculatons to obtan from (see e.g. [ 6]). Another wa to utlze these relatonshps s to combne them wth other avalable equatons. One can then solve these equatons smultaneousl to obtan the desred dstrbutons. To show how t wors we consder a stable D-BMAP/G/ queue under the LAS-DA as an eample. Let S be an ndependent and dentcall dstrbuted.

14 random varable of servce tme wth Pr( S = ) = s. Applng the so-called arrval tme approach (Chae et al. [5]) one can obtan the followng equatons: ss Y( = ( ρ ) g + ρx ( A ( () E where ss X ( = X ( A( z (2) ρ = λe( S) > 0 whch represents the probablt that the server s bus at an arbtrar slot g s the probablt vector that represents the dstrbuton of the phase of the UMC when the server s dle. (It can be shown that g s the nvarant probablt vector of G.e. gg = g and ge = where G represents the transton probablt matr of the phase change durng the so-called fundamental perod (Neuts [0 p.6]). Ths s a dscretetme analogue of the contnuous-tme BMAP result (Lucanton [9]).) ss X ( s the generatng functon of the jont probablt vector of the number of customers n the sstem and the phase of the UMC just after a servce start epoch A ( = s ( ) = D z whch represents the matr generatng functon of the jont probablt of the number of customers arrvng durng a servce tme and the phase change of the UMC durng the servce tme and A ( E represents the matr generatng functon of the jont probablt of the number of customers arrvng durng elapsed slots snce the begnnng of the ongong servce and the phase change of the UMC durng these elapsed slots. It s eas to show that A (z E ) has the followng relaton to A ( : A ( ( I D( ) = E( S) ( I A( ). E Note that () s obtaned b condtonng on Y ( b whether the server s bus or not: n partcular under the condton that the server s bus at an arbtrar slot the number of customers n the sstem at ths slot equals the number at the begnnng of the current servce plus the number of customers arrvng durng the elapsed slots snce the

15 begnnng of ths servce. Then consderng the phase change durng these elapsed slots () follows. Also note that (2) s obtaned from the obvous relaton that the number of customers just after a servce completon epoch equals the number at the begnnng of the servce plus the number of customers arrvng durng the servce tme mnus one (for the departng customer). For detals see [5]. Now smultaneousl solvng () (2) and Y( ( D( I) = λ ( z ) X( (see (7)) we fnall have Y( ( A( zi) = ( ρ )( ga( X( ( A( zi) = λ ( ρ) ga( ( I D( ) X ss ( ( A( zi) = λ ( ρ) zg( I D( ). (See e.g. [9] for the contnuous counterparts of these results.) Y ( and Y ( then follow from Corollares 2 and 3. We hope that our ntutve dervatons and the results obtaned would help readers better understand the dscrete-tme queues wth D-BMAP arrvals.

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