SOJOURN TIME IN A QUEUE WITH CLUSTERED PERIODIC ARRIVALS

Transcription

1 Journal of the Operatons Research Socety of Japan 2003, Vol. 46, No. 2, he Operatons Research Socety of Japan SOJOURN IME IN A QUEUE WIH CLUSERED PERIODIC ARRIVALS Da Inoue he oko Marne and Fre Insurance Co., Ltd etsuya akne Kyoto Unversty Receved Receved May 27, 2002; Revsed December 24, 2002) Abstract hs paper consders a FIFO sngle-server queue wth ndependent and homogeneous sources. Each source generates exactly one message every nterval of fxed length. Each message s then dvded nto a constant number of fxed-sze cells and these cells arrve to the queue back to back as f they form a tran of cells. We call ths arrval process clustered perodc arrvals. he queue wth clustered perodc arrvals s an obvous generalzaton of D/D/1 queue whch corresponds to the case that each message conssts of only one cell. hs paper derves the statonary probablty dstrbutons of sojourn tmes of respectve cells n a message. An nterestng feature of these sojourn tme dstrbutons s that they are not contnuous functons of tme, but they have masses at multples of the cell transmsson tme. hs paper also derves the jont probablty dstrbuton of dfferences between sojourn tmes of successve cells n a message and the mean watng tmes of respectve cells n a message. At last, the overall mean watng tme n the queue wth clustered perodc arrvals s compared wth those n the correspondng queues wth dspersed perodc arrvals and perodc batch arrvals, and the effcency of dspersng cells s quanttatvely shown by smple formulas. Keywords: Queue, telecommuncaton, sojourn tme, clustered perodc arrval 1. Introducton AM s consdered as one of the most promsng transfer technologes for mplementng Broadband ISDN that provdes servce to such dverse traffc as vdeo, stll mage, voce and data. In AM, nformaton flow s organzed nto fxed-sze cells. CBR s one of servce classes supported by AM, where CBR sources generate bt streams at a constant rate. For example, vdeo wth fxed rate encoders and codng voce result n CBR traffc streams. hus the performance of multplexers wth CBR traffc streams has been studed extensvely. So far, queues wth the superposton of perodc arrval processes have been studed n a large number of research papers see [9] and references theren). In partcular, an algorthmc soluton to the complementary dstrbuton of queue length n the D/D/1 queue was derved n [6], where all sources are ndependent and homogeneous and the number of sources s fxed. In [5] and [16], the steady-state delay dstrbutons were derved n a dscrete-tme and contnuous-tme D/D/1 queues, respectvely. In [10], an analytcal approach usng the Ballot heorems was shown to obtan steady-state delay dstrbutons both n a dscrete-tme and contnuous-tme D/D/1 queues. Note that n those papers, each source s assumed to generate cells one by one every tme nterval of fxed length. Accordng to a recent report on montorng of CBR traffc generated by a CBR MPEG encoder, the cell stream generated by the CBR MPEG encoder turns not to be a CBR n cell-level, but t s a clustered cell stream at a constant rate [13]. More precsely, the CBR MPEG encoder generates eght cells correspondng to the sze of PDU of the upper layer) 220

2 Queue wth Clustered Perodc Arrvals 221 generaton epoch [second] cell number Fgure 1: Cell generaton epochs from a CBR MPEG encoder. back to back at full rate every nterval of fxed length see Fgure 1). Such a feature may be found n cell-level traffc of other streamng applcatons. In ths paper, we call clustered cells correspondng to a PDU of the upper layer a message and the arrval pattern n cell level clustered perodc arrvals. For ths knd of arrvals, of partcular nterest s the length of tme from a message generaton epoch.e., a generaton epoch of the frst cell n a message) to the end of the last cell s transmsson n the message. Cdon et al. [4] found the statonary probablty dstrbutons of unfnshed work for queues wth dscrete-tme clustered perodc arrvals and wth flud clustered perodc arrvals. Note that from those results, we can obtan only the statonary probablty dstrbuton of sojourn tmes of the frst cells n messages or the frst part of flud messages). Related work s also found n [1], [2], [3], [7], [8], [9], [12], [15] and [17], where each source s allowed to generate more than one cell every tme nterval of fxed length. We consder a contnuous-tme sngle-server FIFO queue wth ndependent and homogeneous clustered perodc arrvals. Note that our model s not a flud model but a model wth nstantaneous arrvals, and therefore t s an obvous generalzaton of D/D/1 queue whch corresponds to the case that each message conssts of only one cell. We obtan explct formulas for the probablty dstrbutons of sojourn tmes of respectve cells n a message. he message-level performance, whch s mportant n dmensonng network resources, s then obtaned by addng the constant lag.e., tme nterval between generaton epochs of the frst and last cells n a message) to the sojourn tme of the last cell. As by-products, the statonary probablty dstrbutons of the amount of unfnshed work and the queue length are also obtaned. An nterestng feature of the sojourn tme dstrbutons of respectve cells n a message s that they are not contnuous functons of tme, but they have masses at multples of the cell transmsson tme. Moreover we obtan the jont probablty dstrbuton of dfferences between sojourn tmes of successve cells n a message, and usng ths result we derve the explct expressons for the mean watng tmes of respectve cells. Fnally, We wll compare the overall mean watng tme of the queue wth clustered perodc arrvals wth those n the correspondng queues wth dspersed perodc arrvals [9] and perodc batch arrvals, and the effcency of dspersng cells s quanttatvely shown by smple formulas. he rest of ths paper s organzed as follows. In secton 2, we descrbe the mathematcal model. In secton 3, we derves the probablty dstrbutons of sojourn tmes of respectve cells. In secton 4, we obtan the jont probablty generatng functon of the dstrbutons c Operatons Research Socety of Japan JORSJ 2003) 46-2

3 222 D. Inoue &. akne τ 0 τ 0 τ 0 + M Source 0 τ 1 Source 1 Source K M + 1 Perod τ K Fgure 2: Orgnal system. of dfferences of sojourn tmes of successve cells. Fnally, n secton 5, we derve the overall mean watng tme and compare t wth those n the correspondng systems wth dspersed perodc arrvals and perodc batch arrvals. 2. Model and Equvalent Systems 2.1. Model descrpton We consder a work-conservng sngle-server queue fed by K + 1 K 0) ndependent and homogeneous sources labeled 0 to K. Each source generates messages of fxed length perodcally wth perod of length. We assume that source j j = 0,..., K) generates messages at tme τ j, τ j ±, τ j ±2,..., where τ j s are ndependent and dentcally dstrbuted accordng to a unform dstrbuton n nterval [0, ). Each message s then dvded nto M + 1 M 0) cells of fxed length. When a message s generated at tme t, the th = 1,..., M + 1) cell n the message arrves to the queue at tme t + 1 see Fgure 2). he queue has a buffer of nfnte capacty, so that no cell s lost. Cells are served by a sngle server on an FIFO bass. Servce tmes of cells are constant and are chosen as a unt of tme. Because the amount of work brought nto the system durng any nterval of length s equal to K + 1)M + 1), we assume that K + 1)M + 1), 1) so that the system s stable. Remark 1 When the queue s stable, the maxmum queue length s equal to M + 1)K + 1 cells, so that no cells are lost f the capacty of the buffer s not less than M + 1)K + 1 cells. We defne U t as the amount of unfnshed work at tme t. he amount U t of unfnshed work s decreasng at rate 1 whle U t > 0 and U t has an upward jump by 1 when a cell arrves. We assume that U t s rght-contnuous and has left-hand-sde lmts. hus when a cell arrves at tme t, U t ncludes the servce tme of ths cell, so that U t 1. Let Ax, y], A[x, y] and Ax, y) be the number of cells arrvng n nterval x, y], [x, y] and x, y), respectvely. We now assume that the queueng process U t begns at tme τ τ > 0) wth U τ = 0. We then have [4] U t = max {At u, t] u}, t τ. 2) 0 u t+τ Proposton 1 Suppose the stablty condton 1) holds and U τ = 0. hen U t+ = U t for all t τ +. c Operatons Research Socety of Japan JORSJ 2003) 46-2

4 Queue wth Clustered Perodc Arrvals 223 Proof. Substtutng t n 2) by t +, we obtan for all t τ +, U t+ = max At + u, t + ] u) 0 u t+τ + { } = max max 0 u t+τ At + u, t + ] u), max At + u, t + ] u) t+τ u t+τ + { } = max max 0 u t+τ At u, t] u), max At t+τ u t+τ u, t + ] u ), 3) where we use the perodcty At u, t] = At + u, t + ] n the last equalty. Note here that max At t+τ u t+τ u, t + ] u ) = max At u, t] u + At, t + ] ) t+τ u t+τ max t+τ u t+τ At u, t] u) max 0 u t+τ At u, t] u), where the frst nequalty follows from the stablty condton At, t+ ] = K+1)M+1) see 1)). It then follows from 3) that U t+ = max At u, t] u) = U t, 0 u t+τ for all t τ +, whch completes the proof. Because K + 1 sources are ndependent and homogeneous, we focus on the sojourn tme dstrbuton of the + 1)st = 0,..., M) cells n messages generated by source 0. o do so, we shft the tme axs by τ 0 n such a way that source 0 generates messages at tme 0, ±, ±2,.... Further, we assume that U τ = 0 for some τ 2 n the shfted tme axs. It s easy to see from Proposton 1 that the system s perodc after epoch τ + ) when the stablty condton 1) holds. hus the sojourn tmes for the + 1)st cells s equvalent to the amount U of unfnshed work at tme because cells are served on a FIFO bass and U t s assumed to be rght-contnuous Equvalent systems o obtan the probablty dstrbuton of the amount U = 0,..., M) of unfnshed work at tme n the orgnal system, we follow an dea n [4]. Namely, we ntroduce an equvalent system for the +1)st cells = 0,..., M) whch s also called system E n short. In system E = 0,..., M), nothng s changed except that all cells n messages generated n nterval τ, M] n the orgnal system arrve at the same tme when the frst cells arrve,.e., batch arrvals see Fgure 3). Note that arrval epochs of cells n messages generated after tme M n system E are dentcal to those n the orgnal system. We defne U ) t = 0,..., M) as the amount of unfnshed work at tme t n the equvalent system E for the + 1)st cells. Let A ) x, y], A ) [x, y] and A ) x, y) denote the number of cells arrvng n nterval x, y], [x, y] and x, y), respectvely, n system E. We then have U ) t = max 0 u t+τ A ) t u, t] u ), t τ. 4) Note here that A ) τ, t] = A τ, t], t. 5) c Operatons Research Socety of Japan JORSJ 2003) 46-2

5 224 D. Inoue &. akne M 0 M Source 0 Source 1 Source K M + 1 Perod Fgure 3: System E : Equvalent system for the + 1)st cells. Proposton 2 he amount U ) = 0,..., M) of unfnshed work at tme n the equvalent system for the + 1)st cells s dentcal to the amount U of unfnshed work at tme n the orgnal system. Proof. o prove the theorem, we frst consder the correspondng system wth batch arrvals, where t s empty at tme τ, all messages are generated at the same tme as n the orgnal system and M+1 cells n each message smultaneously arrve to the system when the message s generated. Let Cx, y] denote the number of messages generated n nterval x, y]. We defne U B) t as the amount of unfnshed work at tme t n the correspondng system wth batch arrvals. We then have U B) t = max 0 u t+τ M + 1)Ct u, t] u), t τ. 6) Note that U ) t = U B) t = 0,..., M) for t [ τ, M] because A ) t u, t] = M + 1)Ct u, t] for all t τ, M] and u 0, t + τ ], and n general, U t U ) t U B) t, t τ, = 0,..., M, 7) because At u, t] A ) t u, t] M + 1)Ct u, t] for all t > τ and u 0, t + τ ]. It s easy to see from 7) that f U B) t = 0, U t = U ) t = 0. Conversely, f U t = 0, ths mples that all messages generated n τ, t] have been already served before tme t, so that U ) t = U B) t = 0. hus we have the followng lemma. Lemma 1 Both the orgnal system and the equvalent system E for the + 1)st cells are busy f and only f U B) t > 0. When the stablty condton 1) holds, t s easy to see that there exsts a tme nstant t, at whch U B) t = 0 and U B) t = M + 1 > 0, n any nterval of length begnnng after tme τ + ). Because a message from source 0 s generated at tme 0, t follows that U B) t > 0 for all t [0, ]. hus there exsts a tme nstant t [, 0) such that U B) t = 0 and U τ B) > 0 for all τ [t, ]. hus t follows from Lemma 1 that both the orgnal system and system E are busy n [t, ]. Further, Lemma 1 or 7)) mples that U t = U ) t = 0. herefore we obtan U = A[t, ] t ), 8) U ) = A ) [t, ] t ). 9) c Operatons Research Socety of Japan JORSJ 2003) 46-2

6 Queue wth Clustered Perodc Arrvals 225 Note here that U t = U ) t = 0 mples that all cells arrvng n nterval τ, t ) have already been served before tme t both n the orgnal system and the equvalent system E for the + 1)st cells. hus A τ, t ) = A ) τ, t ), from whch and 5), t follows that Fnally, wth 8), 9) and 10), we have whch complete the proof. A[t, ] = A ) [t, ]. 10) U = A[t, ] t ) = A ) [t, ] t ) = U ), 3. Sojourn me Dstrbutons In the precedng secton, we showed that sojourn tmes of the + 1)st cells have the same probablty dstrbuton as U ) does. hus, n ths secton, we consder the amount U ) of unfnshed work at tme n the equvalent system for the + 1)st cells to obtan the sojourn tme dstrbuton of the + 1)st cells n the orgnal system Equaton for U ) Let Bx, y] denote the number of messages arrvng n nterval x, y] from sources other than source 0. We defne Gx, y] as the length of an nterval from tme x to the frst generaton epoch of messages n nterval x, y]. Gx, y] = arg mn{q x, y] Bx, q] > 0} x. Let Gx, y] = y x f Bx, y] = 0. heorem 1 U ) = 0,..., M) s gven by { } U ) = max G M, 0], max M + 1)B M u, M] u) 0 u M Proof. Settng t = n 4) yelds U ) M + kb k, k + 1] + M ) = max A ) u, ] u ) 0 u +τ { = max max 0 u A ) u, ] u ), max A ) u, ] u )}. 12) u +τ Note here that τ 2 and accordng to the same argument as n the proof of Proposton 1, max A ) u, ] u ) u +τ = max 0 u +τ = max 0 u +τ max { = max 0 u +τ max 0 u A ) u, ] u ) A ) u, u] + A ) u, ] u ) A ) u, ] u ) A ) u, ] u ), max A ) u, ] u )}. u +τ c Operatons Research Socety of Japan JORSJ 2003) 46-2

7 226 D. Inoue &. akne hus when + τ = k + x for some postve nteger k and 0 x <, we repeat the above procedure and obtan max A ) u, ] u ) { max max A ) u, ] u ), max A ) u, ] u )} u +τ 0 u 0 u x = max A ) u, ] u ). 13) 0 u herefore we obtan from 12) and 13) U ) { = max A ) u, ] u } 0 u { = max max A ) u, ] u ), max 0 u M { = max A ) u, ] u ), max 0 u M M u A ) u, ] u )} max A ) M u, M] u + A ) M, ] M )}. 14) 0 u M We now consder the frst component max 0 u M A ) u, ] u ) on the rght hand sde of 14). Let g ) u) = A ) u, ] u for 0 u M. Suppose cells arrve at t j j = 0,..., m) durng nterval M, ], where t j 1 < t j j = 1,..., m). Note that t 0 = 0 and t m = M G M, 0]. Because M +1 cells arrve one by one wth ntervals of length one once a message s generated, we have t j t j 1 1 for all j = 1,..., m. Further each cell brngs one unt of unfnshed work nto the system. hus the sequence g ) t 0 +), g ) t 1 +),..., g ) t m +) s nondecreasng, whle g ) u) s a locally decreasng functon of u n nterval t j 1, t j ] j = 1,..., m). Further g ) u) s a decreasng functon of u n nterval t m, M]. As a result, the frst component n 14) attans ts maxmum at u = M G M, 0])+. hus we have max 0 u M A ) u, ] u ) = A ) [ M + G M, 0], ] M G M, 0]) = A ) M, ] M G M, 0]), 15) where the second equalty follows from the fact that there are no arrvals n M, M + G M, 0]). Note also that A ) M u, M] = M + 1)B M u, M], u [0, M + τ ], 16) M A ) M, ] = kb k, k + 1] + + 1), 17) where cells arrvng from source 0 contrbute to the last term, + 1, n the second equaton. hus t follows from 14), 15), 16) and 17) that U ) { M ) = max kb k, k + 1] + + 1) M G M, 0]), M max kb k, k + 1] + + 1) M 0 u M + M + 1)B M u, M] u) }, whch s equvalent to 11). c Operatons Research Socety of Japan JORSJ 2003) 46-2

8 Queue wth Clustered Perodc Arrvals 227 M Source 0 0 B, M] = k 1 B M, 0] = k 2 B0, ] = k 3 G M, 0] Fgure 4: Events I 1 k 1 ), I 2 k 2 ) and I 3 k 3 ) Dstrbuton functon of U ) In ths subsecton, we derve the dstrbuton functon of the amount U ) of unfnshed work at tme n the equvalent system for the + 1)st cells. Let I 1 k 1 ), I 2 k 2 ) and I 3 k 3 ) denote the events B, M] = k 1, B M, 0] = k 2 and B0, ] = k 3, respectvely. See Fgure 4. By defnton, k 1 + k 2 + k 3 = B, ] = K, B M, 0] = k 2 = 0 for = M, and B0, ] = k 3 = 0 for = 0. We frst consder the dstrbuton functon of U ) by condtonng those three events. hat s, Pr U ) v ) = k 1 +k 2 +k 3 =K k 1,k 2,k 3 0 PrI 1 k 1 ), I 2 k 2 ), I 3 k 3 )) Pr U ) v I 1 k 1 ), I 2 k 2 ), I 3 k 3 ) ), where PrI 1 k 1 ), I 2 k 2 ), I 3 k 3 )) follows a multnomal dstrbuton: PrI 1 k 1 ), I 2 k 2 ), I 3 k 3 )) = K! k 1!k 2!k 3! ) M k1 ) M k2 ) k3, because message generaton epochs of K sources are ndependent and dentcally dstrbuted accordng to a unform dstrbuton over any nterval of length. It follows from 11) that the event U ) v happens f and only f M G M, 0] + kb k, k + 1] + M + 1 v, max M + 1)B M u, M] u) + M kb k, k + 1] + M + 1 v, 0 u M or equvalently, M kb k, k + 1] j ) v G M, 0]), max M + 1)B M u, M] u) v + M 1 M kb k, k + 1], 0 u M 19) where j ) x) = x + M 1, 20) and x represents the maxmum nteger whch s not greater than x. 18) c Operatons Research Socety of Japan JORSJ 2003) 46-2

9 228 D. Inoue &. akne We defne q ) j k 2, k 3, w) and R ) y k 1 ) as q ) j k 2, k 3, w) M ) = Pr kb k, k + 1] = j I 2 k 2 ), I 3 k 3 ), G M, 0] = w, 21) ) R ) y k 1 ) = Pr max M + 1)B M u, M] u) y I 1 k 1 ). 22) 0 u M Note here that M ) Pr kb k, k + 1] = j I 1k 1 ), I 2 k 2 ), I 3 k 3 ), G M, 0] = w M ) = Pr kb k, k + 1] = j I 2k 2 ), I 3 k 3 ), G M, 0] = w = q ) j k 2, k 3, w), ) Pr max M + 1)B M u, M] u) y I 1 k 1 ), I 2 k 2 ), I 3 k 3 ) 0 u M ) = Pr max M + 1)B M u, M] u) y I 1 k 1 ) 0 u M = R ) y k 1 ). hus the two probabltes q ) j k 2, k 3, w) and R ) y k 1 ) are condtonally ndependent gven I j k j ) j = 1, 2, 3) and G M, 0] = w. Note here that 22) s rewrtten to be R ) y k 1 ) = PrM + 1)B M u, M] u y, u [0, M] I 1 k 1 )). 23) In what follows, an empty sum s defned as zero. Lemma 2 R ) y k 1 ) s gven by R ) y k 1 ) = 1 k 1 j= y M+1 +1 k1 he proof s gven n Appendx A. M) M + 1)k 1 + y M) M + 1)j + y ) ) j M + 1)j y 1 j M ) k1 j M + 1)j y. 24) M Remark 2 Consder the specal case of M = 0 and k 1 = K for = 0 n 24). R 0) y K) = K j= y +1 + y K + y j K j ) j ) y j 1 j y ) K j. 25) Note that R 0) y K) for M = 0 represents the probablty dstrbuton functon of the amount of unfnshed work n the conventonal D/D/1 queue wth K sources of perod, whch s consdered n secton 3-C of [10]. Conversely, 24) s obtaned by substtutng y/m + 1), k 1 and M)/M + 1) for y, K and, respectvely, n 25). hus 24) tself s consdered as the the dstrbuton functon of the amount of unfnshed work n the conventonal D/D/1 queue wth a partcular settng of parameters. c Operatons Research Socety of Japan JORSJ 2003) 46-2

10 Queue wth Clustered Perodc Arrvals 229 In what follows, we derve the condtonal dstrbuton PrU ) v I 1 k 1 ), I 2 k 2 ), I 3 k 3 )) by consderng two cases, k 2 = 0 and k 2 1, separately. Lemma 3 PrU ) x I 1 k 1 ), I 2 0), I 3 k 3 )) s gven by PrU ) v 1 v I 1 k 1 ), I 2 0), I 3 k 3 )) = q ) j 0, k 3, M )R ) v+m 1 j k 1 ). 26) j=0 Proof. Note frst that when the event I 2 0) happens, B M, 0] = 0, so that the frst arrval n M, 0] occurs from source 0 at tme 0. hus G M, 0] = M. It then follows from 19) that Pr U ) v M = Pr = I 1 k 1 ), I 2 0), I 3 k 3 ) ) kb k, k + 1] j ) v M )), j ) v M+) j=0 max M + 1)B M u, M] u) 0 u M M v + M 1 q ) j 0, k 3, M ) Pr ) kb k, k + 1] I 1k 1 ), I 2 0), I 3 k 3 ) max 0 u M hus 26) follows from 20), 22) and the above equaton. Lemma 4 For k 2 1, PrU ) v I 1 k 1 ), I 2 k 2 ), I 3 k 3 )) s gven by Pr U ) v I 1 k 1 ), I 2 k 2 ), I 3 k 3 ) ) = M 1 l=0 k 2 k 2! M l 1) k 2 k 4 k 4!k 2 k 4 )! M ) k 2 k 4 =1 { v l+m 2 j=k 4 M l) M + 1)B M u, M] u) ) v + M 1 j I 1k 1 ). q ) j k 4 M l) k 2 k 4, k 3, l + 1, ) R ) v + M 1 j k 1 ) + [ 1 1 v + v ) k 4] q ) v k 4 1)M l) 1 k 2 k 4, k 3, l + 1) } R ) v v + l k 1 ). 27) Proof. Let fy k 2 ) denote the condtonal probablty densty functon of G M, 0] gven I 2 k 2 ) for k 2 1. Because k 2 arrvals are ndependent and unformly dstrbuted over the nterval M, 0], we have o obtan PrU ) fy k 2 ) = k 2 M ) M y k2 1, 0 y M. M v I 1 k 1 ), I 2 k 2 ), I 3 k 3 )), let I 4 k 4 ) denote the followng event: B M + G M, 0], M + G M, 0] ] = k 4 1, c Operatons Research Socety of Japan JORSJ 2003) 46-2

11 230 D. Inoue &. akne B M + l, M + l + 1] = k 4 Source 0 M 0 l x B, M] = k G M, 0] 1 B0, ] = k 3 B M + l + 1, 0] = k 2 k 4 Fgure 5: Events I k ) = 1,..., 4) n the case k 2 1. where y represents the mnmum nteger whch s not smaller than y. Suppose the nterval M, 0] s dvded nto M slots of length one. hen the event I 4 k 4 ) mples that k4 messages arrve n the frst slot among those havng message arrvals. hus, gven the event I 2 k 2 ), we have 1 k 4 k 2, and the number of messages arrvng n the nterval from the end of the frst slot wth message arrvals to tme 0 s gven by k 2 k 4. See Fgure 5. We now dvde G M, 0] nto the nteger part l and the decmal part x,.e., G M, 0] = l + x wth 0 x < 1. hen, usng 19) and condtonng events G M, 0] = l + x and I 4 k 4 ), we obtan Pr U ) v I1 k 1 ), I 2 k 2 ), I 3 k 3 ) ) = M 1 l=0 1 0 Pr dxfl + x k 2 ) k 4 M l) + max 0 u M k 2 k 4 =1 M l 1 Pr I 4 k 4 ) I 2 k 2 ), G M, 0] = l + x) kb k, k + 1] j ) v l + x)), M + 1)B M u, M] u) M l 1 v + M 1 k 4 M l) kb k, k + 1] ) I 1 k 1 ), I 2 k 2 ), I 3 k 3 ), I 4 k 4 ), G M, 0] = l + x. Further, by condtonng the value of M l 1 kb k, k + 1] and usng 21) and 22), we rewrte the above equaton to be Pr U ) = M 1 l=0 v I 1 k 1 ), I 2 k 2 ), I 3 k 3 ) ) 1 0 j ) v l+x)) j=k 4 M l) Note here that dxfl + x k 2 ) k 2 k 4 =1 Pr I 4 k 4 ) I 2 k 2 ), G M, 0] = l + x) q ) j k 4 M l) k 2 k 4, k 3, l + 1) R ) v + M 1 j k 1 ). 28) Pr I 4 k 4 ) I 2 k 2 ), G M, 0] = l + x) = PrB M + l + x, M + l + x ] = k 4 1 B M + l + x, 0] = k 2 1) ) k4 1 k 2 1)! x x ) k2 k M l x 4 =, k 4 1)!k 2 k 4 )! M l x M l x c Operatons Research Socety of Japan JORSJ 2003) 46-2

12 Queue wth Clustered Perodc Arrvals 231 and j ) j ) v l), v l x) = j ) v l 1), x v v, x > v v. We defne x ) v) as hen 28) s rewrtten to be Pr U ) = = v I 1 k 1 ), I 2 k 2 ), I 3 k 3 ) ) x ) v) ) 1 + dx k 2 0 x ) v) M M 1 l=0 M 1 l=0 k 2 k 4 =1 k 2 k 4 =1 k 2 1)! k 4 1)!k 2 k 4 )! j ) v l x) j=k 4 M l) x ) v) = v v. 29) x x M ) k4 1 M l x M ) k2 k 4 q ) j k 4 M l) k 2 k 4, k 3, l + 1) R ) v + M 1 j k 1 ) k 2! M l 1) k 2 k 4 k 4!k 2 k 4 )! M ) k 2 { [1 ] j) v l) 1 x ) v)) k 4 j=k 4 M l) R ) v + M 1 j k 1 ) + 1 x ) v)) k 4 j ) v l 1) j=k 4 M l) q ) j k 4 M l) k 2 k 4, k 3, l + 1) q ) j k 4 M l) k 2 k 4, k 3, l + 1) R ) v + M 1 j k 1 ) hus 27) follows from 20), 29) and the above equaton. Fnally, usng 18), Lemmas 3 and 4, we obtan the followng theorem. heorem 2 Let S = 0,..., M) denote a generc random varable representng sojourn tmes of the + 1)st cells. We then have }. PrS v) = k 1 +k 2 +k 3 +k 4 =K k 1,k 2,k 3 0,k 4 1 M 1 l=0 K! M) k 1 k 1!k 2!k 3!k 4! K { v l+m 2 j=k 4 M l) n ) j k 4 M l) k 2, k 3, l + 1) R ) v + M 1 j k 1 ) [ 1 1 v + v ) k ] 4 n ) v k 4 1)M l) 1 k 2, k 3, l + 1) } R ) v v + l k 1 ) + v 1 j=0 k 1 +k 3 =K k 1,k 3 0 K! M) k 1 k 1!k 3! K n ) j 0, k 3, M )R ) v + M 1 j k 1 ), 30) c Operatons Research Socety of Japan JORSJ 2003) 46-2

13 232 D. Inoue &. akne where R ) y k 1 ) s gven n 24), n ) j k 2, k 3, l) s defned as and they are recursvely computed by n ) j k 2, k 3, l) = M l) k 2 k 3 q ) j k 2, k 3, l), 31) n ) j 0, 0, l) = n ) j k 2 + 1, k 3, l), = n ) j k 2, k 3 + 1, l) = { 1 j = 0) 0 j 0), 32) M l m=+1 m=1 n ) j m k 2, k 3, l), 33) n ) j m k 2, k 3, l). 34) Proof. Recall that PrS v) = PrU ) v). hus we obtan 30) by substtutng 26) and 27) nto 18), usng 31) and notng wth k 2 = k 2 k 4 k 1 +k 2 +k 3 =K k 1,k 3 0,k 2 1 K! k 1!k 2!k 3! k 2 k 4 =1 k 2! k 4!k 2 k 4 )! = k 1 +k 2 +k 3+k 4 =K k 1,k 2,k 3 0,k 4 1 K! k 1!k 2!k 3!k 4!. On the other hand, from 21) and 31), n ) j k 2, k 3, l) can be regarded as the number of ways such that M l kb k, k + 1] s equal to j gven that three events I 2k 2 ), I 3 k 3 ) and G M, 0] = l occur. Wth ths observaton, we have 32)-34). From heorem 2, we can derve the probablty dstrbutons of the amount of unfnshed work and the queue length. Note that the frst cells n respectve messages from source 0 see the tme average of unfnshed work n the system wth source 0 removed [18]. On the other hand, the queue length s unquely determned by the amount of unfnshed work because servce tmes of all cells are equal to one. hus we have the followng corollary. Corollary 1 For the system wth K sources, the probablty dstrbuton functon of the amount of the statonary unfnshed work s gven by PrS 0 v), and the statonary queue length dstrbuton PrL k) s gven by PrS 0 k + 1) k = 0, 1,..., M + 1)K 1) + 1), where L denotes a generc random varable representng the statonary queue length. Remark 3 Consder the specal case of M = 0.e., the conventonal D/D/1) and PrS 0 v). In ths specal case, the frst term on the rght hand sde of 30) vanshes. Further n the summaton n the second term, only the term for k 1 = K and j = 0 remans because PrI 3 0)) = 1 and n j) 0, 0, 0) = 0 for all j 1. hus PrS 0 v) n 30) s reduced to R 0) v 1 K). hus, the sojourn tme s gven by the sum of the amount of unfnshed work n the system wth K sources and the servce tme n ths specal case. For K + 1 = 8, = 128 and M + 1 = 8, we plot the dstrbuton functon PrS v) and ts complement PrS > v) n Fgure 6 a) and b), respectvely. From Fgure 6 a), we observe that PrS v) = 1,..., M) s a dscontnuous functon of v. hs s due to the fact that sojourn tmes of all cells n a message whch starts a busy perod always take nteger values. hus the sojourn tme dstrbutons of the second to the last cells have masses at ntegers. Further, from Fgure 6 b), we observe that the formula n heorem 2 s numercally stable. c Operatons Research Socety of Japan JORSJ 2003) 46-2

14 Queue wth Clustered Perodc Arrvals 233 PrS <v) S S 1 S S 3 S S 5 S S 7 v logprs >v) S 7 S 6 S 5 S 4 S 3 S 2 S 1 S 0 a) Probablty dstrbuton of S v b) Complementary dstrbuton of S n log scale Fgure 6: Sojourn tme dstrbutons K + 1 = 8, = 128 and M + 1 = 8). 4. Jont Dstrbuton of Dfferences of Sojourn mes In ths secton, we consder the jont dstrbuton of the dfferences of sojourn tmes S = 0,..., M) of the + 1)st cells n a message. Recall that S = U. hus t follows from 2) that S = S 1 + A 1, ] 1, = 1,..., M, because U t > 0 for all t [0, M]. Let = 1,..., M) denote the dfference between sojourn tmes of the th and 1)st cells n the message generated at tme 0: We then have = S S 1, = 1,..., M. = A 1, ] 1 = B M 1, ] + 1) 1 = B M 1, ]. 35) hus = 1,..., M) takes nteger values and 0 K. We defne P z 1,..., z M ) as the probablty generatng functon of the jont dstrbuton of = 0,..., M). P z 1,..., z M ) = E [ ] z 1 1 z M M. c Operatons Research Socety of Japan JORSJ 2003) 46-2

15 234 D. Inoue &. akne heorem 3 P z 1,..., z M ) s gven by P z 1,..., z M ) = 2M + 1 M j j=1 l=1 z l + 1 K M M z l. 36) j=1 l=j Proof. It follows from 35) that hus we have = from whch t follows that P z 1,..., z M ) = E = E 1 k= M 1 z = 0 z Bk 1,k 1 +1] k 1 = M 1 k= M Bk, k + 1], = 1,..., M. 1 k= M 1 1 z Bk,k+1], z 1 z M+1+k ) Bk,k+1] 1 z Bk,k +1] k = M 1 M 1 k=0 M 1 k M = 1 z Bk M,k M +1] M z k+1 z M ) Bk,k+1]. 37) Note here that the probablty generatng functon of the jont dstrbuton of Bk, k + 1] k = M,..., M 1) s gven by E M 1 k= M ω Bk,k+1] k = 2M + M 1 k= M ω k K, 38) because message generaton epochs of sources 1 to K are ndependent and dentcally dstrbuted accordng to a unform dstrbuton over any nterval of length. heorem 3 then follows from 37) and 38). Corollary 2 he probablty mass functon of = 1,..., M) and ts mean are gven by respectvely. ) M ) K + 1 j Pr = j) = 1 M + 1 ) K j, j j = 1,..., K, 39) KM + 1) E[ ] =, 40) Proof. Settng z = z for a specfc and z j = 1 for all j n 36), we have E [ ) ] K z M + 1)z + M 1 =, from whch Corollary 2 follows mmedately. 5. Mean Watng me and Its Comparson to Related Systems In ths secton, we frst derve an explct formula for the mean watng tme of the + 1)st cells. Next, we compare the overall mean watng tme wth those n the correspondng systems wth dspersed perodc arrvals and wth perodc batch arrvals. c Operatons Research Socety of Japan JORSJ 2003) 46-2

16 Queue wth Clustered Perodc Arrvals Mean watng tme Let W = 0,..., M) denote a generc random varable representng watng tmes of the + 1)st cells. Note here that W = S 1, and W = 0 when K = 0. hus we assume K 1 n the rest of ths subsecton. We defne E[W ] as the overall mean watng tme,.e., E[W ] = 1 M + 1 M E[W ]. 41) =0 heorem 4 When K 1, the mean watng tme E[W ] = 0,..., M) of the + 1)st cells s gven by KM + 1) E[W ] = E[W ] + M ), 42) 2 where the overall mean watng tme E[W ] s gven by Proof. From 40), we have E[W ] = M It then follows from 41) and 44) that K 1 j=0 K! j! E[W ] E[W 0 ] = E[S ] E[S 0 ] = E[ j ] E[W ] = 1 M + 1 = E[W 0 ] + = j=1 ) M + 1 K j. 43) M E[W 0 ] + =0 KMM + 1). 2 KM + 1). 44) ) KM + 1) hus substtutng E[W ] KMM + 1)/2 ) for E[W 0 ] n 44) and rearrangng terms yeld 42). o show 43), we employ an nducton wth respect to the number of sources. For k = 1,..., K + 1, let E[W k] = 0,..., M) and E[W k] denote the mean watng tme of the + 1)st cells and the overall mean watng tme, respectvely, n the system wth k sources. Note that, by defnton, E[W ] = E[W K + 1] and E[W ] = E[W K + 1]. We also defne E[U k] k = 1,..., K +1) as the tme-average of the amount of unfnshed work n the system wth k sources. Because of perodcty of U t see Proposton 1), we have E[U K + 1] = 1 t+ t U u du, for any t τ +, where U τ = 0. We frst observe that the sojourn tme of the frst cell n a message generated at tme t from a partcular source s not affected by cells generated before tme t from the same source. Further the arrval epochs of the frst cells are ndependent and dentcally dstrbuted accordng to a unform dstrbuton over any nterval of length, whle the amount of c Operatons Research Socety of Japan JORSJ 2003) 46-2

17 236 D. Inoue &. akne U t x 2 x 1 + 1/2 x 2 + 1/2 x 1 t Perod Fgure 7: Contrbuton of respectve watng tmes to unfnshed work. unfnshed work has a perod of length. hus the frst cell n a message from a partcular source sees the tme average unfnshed work n the system wth ths source removed [18]. We then have E[W 0 k + 1] = E[U k], k = 1,..., K. 45) Next we consder the relatonshp between the amount of unfnshed work and watng tmes of respectve cells. It s easy to see from Fgure 7 that a customer whose watng tme s equal to x contrbutes the amount x + 1) 2 /2 x 2 /2 = x + 1/2 to unfnshed work. hus E[U k] = 1 M ke[w k] + 1 =0 2 ) km + 1) km + 1) = E[W k] +, 46) 2 where we use 41),.e., M =0 E[W k] = M + 1)E[W k]. It then follows from 45) and 46) that km + 1) km + 1) E[W 0 k + 1] = E[W k] +, 2 and therefore from 42) wth = 0 and K = k, we obtan kmm + 1) E[W k + 1] = E[W 0 k + 1] + 2 km + 1) km + 1)2 = E[W k] +, 2 k = 1,..., K. 47) hus 43) s obtaned by a straghtforward nducton wth E[W 1] = Mean watng tme comparson to related systems In ths subsecton, we compare the overall mean watng tme E[W ] to those n the related queueng systems wth K + 1 perodc sources, each of whch generates exactly M + 1 cells n any nterval of length. In partcular, we consder ) the correspondng system wth dspersed perodc arrvals, where successve cells from each source are spread wth an equal dstance /M + 1), and ) the correspondng system wth perodc batch arrvals ntroduced n the proof of Proposton 2. See Fgure 8. Let E [ W D] and E [ W B] denote the overall mean watng tmes n the correspondng systems wth dspersed perodc arrvals and perodc batch arrvals, respectvely. c Operatons Research Socety of Japan JORSJ 2003) 46-2

18 Queue wth Clustered Perodc Arrvals 237 Clustered Perod Dspersed M + 1 Perod /M + 1) Batch M + 1 M + 1 M + 1 Fgure 8: Clustered perodc, dspersed perodc and perodc batch arrvals. heorem 5 E [ W D] and E [ W B] are gven n terms of E[W ]: E [ W D] = E[W ] M + 1, 48) E [ W B] = E[W ] + M 2. 49) Proof. Note that the system wth dspersed perodc arrvals s a specal case of M = 0 and perod /M + 1) n the system wth clustered perodc arrvals. hus substtutng 0 and /M + 1) for M and, respectvely, n 43) yelds E [ W D] = 1 2 K 1 j=0 K! j! ) M + 1 K j. 50) hus 48) mmedately follows from 43) and 50). Next we consder E [ W B]. We defne E [ ] W B = 0,..., M) as the mean watng tme of the + 1)st cells n the correspondng system wth perodc batch arrvals. Note here that E [ ] [ ] W B = E W B 1 + 1, = 1,..., M, from whch t follows that E [ W B] = 1 M + 1 M =0 M E [ ] W B = 1 [ ] ) E W B M =0 = E [ ] W0 B M ) As n the proof of heorem 4, we consder E [ ] W0 B by an nducton wth respect to the number of sources. For k = 1,..., K + 1, let E [ W B k ] and E [ U B k ] = 0,..., M) denote the mean watng tme of the + 1)st cells and the tme-average of the amount of unfnshed work, respectvely, n the correspondng system wth perodc batch arrvals when the system has k sources. Note, by defnton, that E [ W B ] = E [ W B K + 1 ]. Accordng to c Operatons Research Socety of Japan JORSJ 2003) 46-2

19 238 D. Inoue &. akne U t x + M + 1 M + 1)x + M + 1) 2 /2 x Perod t Fgure 9: Contrbuton of respectve watng tmes to unfnshed work. an argument very smlar to that n the proof of heorem 4, we can show that E [ W B 0 k + 1 ] s dentcal to E [ U B k ]. Further, snce each batch arrval whose frst cell s watng tme s equal to x contrbutes the amount x + M + 1) 2 /2 x 2 /2 = M + 1)x + M + 1) 2 /2 to unfnshed work see Fgure 9), we have E [ W B 0 k + 1 ] = E [ U B k ] = k = M + 1)E [ W B 0 k ] + km + 1) E [ W0 B k ] + )/ M + 1)2 2 km + 1)2. 52) 2 Because the recurson 52) for E [ W0 B k ] s the same as 47) for E[W k] and E[W 1] = E [ W0 B 1 ] = 0, we obtan E [ ] W0 B = E[W ], and from 51), 49) mmedately follows. Fnally, we consder the mean watng tmes E[W ] = 0,..., M) and the overall mean watng tme E[W ] n the lmt K whle fxng ρ = M + 1)K + 1)/ < 1. In ths lmt, also goes to nfnty, and the queue wth clustered perodc arrvals converges to the M/D/1 queue wth clustered arrvals, where the sze of a cluster s equal to M + 1. Let E[W ] resp. E[W ] = 0,..., M)) denote the overall mean watng tme resp. the mean watng tme of the + 1)st cells) n the queue wth clustered perodc arrvals n ths lmt. hen E[W ] s obtaned by ether 48) or 49), because the lmts of the mean watng tmes n the correspondng systems are known. For example, the queue wth dspersed perodc arrvals converges to the M/D/1 queue wth arrval rate ρ whose mean watng tme s gven by ρ/{21 ρ)} [11]. Further from heorem 4, we can obtan E[W ]. he results are summarzed n the followng corollary. Corollary 3 Suppose ρ = M + 1)K + 1)/ < 1. hen E[W ] and E[W ] = 0,..., M) are gven by E[W ] = M + 1)ρ 21 ρ), E[W ] = E[W ] + ρ A. Proof of Lemma 2 For a fxed t 1, we defne Ry, t k) as M 2 ), = 0,..., M. Ry, t k) = Pr Bt 1 u, t 1 ] u y, u [0, t] Bt 1 t, t 1 ] = k). c Operatons Research Socety of Japan JORSJ 2003) 46-2

20 Queue wth Clustered Perodc Arrvals 239 Note that Ry, k t) s equvalent to the probablty dstrbuton functon of the amount of unfnshed work n the conventonal D/D/1 queue wth k sources of perod t. Lemma 5 [10]) Ry, t k) s gven by Recall that Ry, t k) = 1 k j= y +1 t k + y t j + y k j ) j ) y j t 1 j y t ) k j. 53) R ) y k 1 ) = PrM + 1)B M u, M] u y, u [0, M] I 1 k 1 )). hus t s easy to see that R ) y k 1 ) = R y M + 1, M M + 1 ) k 1. herefore Lemma 2 mmedately follows from Lemma 5. Because Lemma 5 s provded n [10] wthout proof, we prove t below for completeness. We frst note that Ry, t k) = 1 PrBt 1 u, t 1 ] > u + y, u [0, t] Bt 1 t, t 1 ] = k) = 1 t 0 Prarg max{q [0, t] Bt 1 q, t 1 ] > q + y} = u Bt 1 t, t 1 ] = k)du. Suppose arg max{q [0, t] Bt 1 q, t 1 ] > q + y} = u for some u u 0, t]). hs event s equvalent to ) Bt 1 u, t 1 ] = y + u + δ > y + u, ) Bt 1 u + δ), t 1 ] = y + u + δ, and ) Bt 1 q, t 1 ] q + y for all q [u + δ, t], where δ denotes an nfntesmal postve value. Note here that the condtonal jont probablty of the frst two events s gven by PrBt 1 u, t 1 ] = y + u + δ, Bt 1 u + δ), t 1 ] = y + u + δ) Bt 1 t, t 1 ] = k) = PrBt 1 u + δ), t 1 ] = y + u + δ Bt 1 t, t 1 ] = k) PrBt 1 u + δ), t 1 u] = 0 Bt 1 u + δ), t 1 ] = y + u + δ, Bt 1 t, t 1 ] = k)), and the second term on the rght hand sde of the above equaton converges to one when δ goes to zero. Further Bt 1 q, t 1 ] takes a nonnegatve nteger value and y + 1 Bt 1 u + δ), t 1 ] = y + u + δ k. hus lettng Bt 1 u + δ), t 1 ] = y + u + δ = j, we have Note here that Ry, t k) = 1 k j= y +1 Pr Bt 1 j y), t 1 ] = j Bt 1 t, t 1 ] = k) PrBt 1 q, t 1 ] q + y, q [j y, t] Pr Bt 1 j y), t 1 ] = j Bt 1 t, t 1 ] = k) = herefore 54) s rewrtten to be Ry, t k) = 1 k j= y +1 Bt 1 j y), t 1 ] = j, Bt 1 t, t 1 ] = k). 54) ) j k y j t ) j k j= y +1 1 j y t ) j k y j t ) k j ) j Pr Bt 1 q, t 1 ] q + y, q [j y, t] 1 j y t ) k j. Bt 1 j y), t 1 ] = j, Bt 1 t, t 1 ] = k ). 55) c Operatons Research Socety of Japan JORSJ 2003) 46-2

21 240 D. Inoue &. akne o obtan the expresson of the probablty n the last lne n 55), we defne au) as a functon whch satsfes r 0 au)du = Bt 1 j y) r, t 1 j y)]. Wth au), we have PrBt 1 q, t 1 ] q + y, q [j y, t] Bt 1 j y), t 1 ] = j, Bt 1 t, t 1 ] = k) = PrBt 1 q, t 1 j y)] q + y j q [j y, t] Bt 1 t, t 1 j y)] = k j) r ) t j y) = Pr au)du r r [0, t j y)] au)du = k j. 0 Lemma 6 Ballot heorem [10]) Let au) be an ntegrable perodc real functon on the real numbers, wth perod and values that are ether 0 or greater than or equal to some L > 0. hen, for arbtrary selected s [0, ) s+r ) Pr au)du Lr, r [0, ] au)du = W L = 1 W s 0 L. Substtutng t j y), 1, k j and 0 for, L, W and s, respectvely, n Lemma 6, we obtan Pr Bt 1 q, t 1 ] q + y, q [j y, t] Bt 1 j y), t 1 ] = j, Bt 1 t, t 1 ] = k) = 1 k j t j y) = t k + y t j + y. 56) 53) now follows from 55) and 56). References [1] J. M. Barcelo and J. Garca: Multplexng perodc sources n a tree network of AM multplexers. In Proc. IFIP Broadband Communcatons ) [2] J. M. Barcelo, J. Garca, and O. Casals: Comparson of models for the multplexng of worst case traffc sources. In Proc. IFIP 1st Workshop on AM raffc Management 1995) [3] B. Bensaou, J. Gubert, J. W. Roberts and A. Smonan: Performance of an AM multplexer queue n the flud approxmaton usng the Beneš approach. Annals of Operatons Research, ) [4] I. Cdon, R. Guern, I. Kessler and A. Khamsy: Analyss of a statstcal multplexer wth generalzed perodc sources. Queueng Systems, ) [5] L. G. Dron, G. Ramamurthy and B. Sengupta: Delay analyss of contnuous bt rate traffc over an AM network. IEEE Journal on Selected Areas n Communcatons, ) [6] A. E. Eckberg: he sngle server queue wth perodc arrval process and determnstc servce tmes. IEEE ransactons on Communcatons, ) [7] J. Garca, J. M. Barcelo and O. Casals: An exact model for the multplexng of worst case traffc sources. In Proc. IFIP Conference on Performance of Computer Networks 1995) [8] J. He and K. Sohraby: End-to-end analyss of CBR tandem networks usng ballot theorems. In Proc. IC ) c Operatons Research Socety of Japan JORSJ 2003) 46-2

22 Queue wth Clustered Perodc Arrvals 241 [9] J. He and K. Sohraby: On the queueng analyss of dspersed perodc messages. Proc. IEEE INFOCOM ) [10] P. Humblet, A. Bhargava and M. G. Hluchyj: Ballot theorems appled to the transent analyss of nd/d1 queues. IEEE/ACM ransactons on Networkng, ) [11] L. Klenrock, Queueng Systems, Vol. I: heory John Wley & Sons, New York 1975). [12] K. Kvols and S. Blaabjerg: Bounds and approxmatons for the perodc on/off queue wth applcaton to AM traffc control. In Proc. IEEE INFOCOM ) [13]. Morta, H. atezum, Y. Kawansh, S. Kasahara,. akne and Y. akahash: Smulaton and expermental study on AM mult-node ntegrated connecton. In Proc. the 7th Internatonal Conference on elecommuncaton Systems 1999) [14] A. Prvalov and K. Sohraby: Per-stream jtter analyss n CBR AM multplexors. IEEE/ACM ransactons on Networkng, ) [15] G. Ramamurthy and R. S. Dghe: Dstrbuted source control: A network access control for ntegrated broadband packet networks. IEEE Journal on Selected Areas n Communcatons, ) [16] G. Ramamurthy and B. Sengupta: Delay analyss of a packet voce multplexer by the ΣD /D/1 queue. IEEE ransactons on Communcatons, ) [17] J. W. Roberts, B. Bensaou and Y. Canett: A traffc control framework for hgh speed data transmsson. In Proc. IFIP Workshop on Modellng and Performance Evaluaton of AM echnology 1993) [18] B. Sengupta: A queue wth superposton of arrval streams wth an applcaton to packet voce technology. In Proc. Performance ) etsuya akne Department of Appled Mathematcs and Physcs Graduate School of Informatcs Kyoto Unversty Kyoto , Japan E-mal: takne@amp..kyoto-u.ac.jp c Operatons Research Socety of Japan JORSJ 2003) 46-2