10 9 Channel on time. sec

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1 Blockng of dynamc multcast connectons n a sngle lnk COST 257TD(97)46 Joun Karvo 1, Jorma Vrtamo 2, Samul Aalto 2, Oll Martkanen 1 fjoun.karvo,jorma.vrtamo,samul.aalto,oll.martkaneng@hut.f Helsnk Unversty of Technology 1 Laboratory of Telecommuncaton Software and Multmeda, P.O. Box 1100, FIN HUT 2 Laboratory of Telecommuncatons Technology, P.O. Box 3000, FIN HUT Abstract In ths paper, a method for calculatng blockng experenced by dynamc multcast connectons n a sngle lnk s presented. An orgnatng oce provdes a number of channels dstrbuted to the users by multcast subtrees whch evolve dynamcally as users jon and leave the channels. We reduce ths problem to a generalzed Engset system wth undentcal users and generally dstrbuted holdng tmes, and derve the call and channel blockng probabltes as well as the lnk occupancy dstrbuton. the tree (see gure 1). Ths eect poses a problem when dmensonng a network. Wth multcast trees, less bandwdth s needed near the root of the tree (orgnatng oce) than the sum of the bandwdths used near the leaves. 1 Introducton Call blockng probabltes n a crcut swtched network carryng multple trac classes can be calculated wth exact algorthms, such as the recurson of Kaufman and Roberts [5, 8], or wth approxmatve methods, such as the normal-type approxmaton [7]. These algorthms are applcable for pontto-pont connectons, such as telephone calls or ATM connectons. They apply also for statc multcast connectons, where the structure of each multcast subtree s xed n advance. In a more dynamc envronment, where the subtrees evolve wth arrvng and departng customers, these models are not adequate. Multcast connectons have a bandwdth savng nature. Ths means that a multcast connecton n takng the form of a tree where streams merge at the nodes requres much less capacty from the network lnks than a bunch of separate pont-to-pont connectons from the orgnatng oce to the leaf nodes of Submtted to 4th Internatonal Conference on Broadband Communcatons '98, Stuttgart, Germany Correspondng author: Joun Karvo, researcher at Helsnk Unversty of Technology, P.O. Box 1100, FIN HUT, tel: , fax: , e-mal: joun.karvo@hut.f Fgure 1: Pont-to-pont (top) vs. pont-tomultpont, or multcast connectons (bottom). Data streams are presented wth a thck broken lne. Dot et al. [2] gve a revew on the work n the area of multcast trac. Most work n the sense of blockng n ATM networks wth multcastng has been made on blockng n multcastng-capable swtches, see, for example, [3] or [6]. Shacham and Yokota [10] propose call admsson control algorthms for realtme multcast transmsson. We use a settng comparable to ther problem and devce an algorthm for calculatng call blockng probabltes. The model of dynamc multcast connectons used n ths paper pertans, e.g., to the case n whch TV or rado staton provdes several programs to vewers Page 1

2 or lsteners va a telecommuncaton network. The model conssts of a sngle multcast tree, whose root s called the orgnatng oce and leaves are called users. The orgnatng oce oers the users a set of programs, called channels. The programs run ndependently of ther subscrbers, who can jon and leave the channel any tme. The connectons that delver a specc channel to a group of users form a dynamc subtree of the multcast tree. A jonng user s assumed to choose a channel probablstcally accordng to a channel preference dstrbuton, whch s the same for all the users. When jonng, the user creates a new branch to the subtree extendng from the leaf to the nearest node, A, already connected to the channel (see gure 2). Blockng may occur on any lnk of the new branch. On the other hand, there s no blockng on the lnks upstream from the connectng node A, snce on that path the channel s already on. Note, however, that the jonng user may extend the tme the channel remans swtched on. O A Fgure 2: The subtree of a channel wth a new branch created by a jonng user U. Several multcast subtrees may use the same lnk. The requred capacty n the lnks vares accordngly, sometmes leadng to blockng when the requested capacty s not avalable. The probablty that a channel does not nd sucent capacty on a specc lnk s called channel blockng probablty on that lnk. By the call blockng probablty we mean the probablty that the user's request to subscrbe to a channel s blocked because of the channel blockng occurng on U some of the lnks between the leaf and the connectng node A. In ths paper we show how to calculate the blockng probabltes of dynamc multcast connectons n a specc lnk wth nte capacty assumng that all the other lnks have nnte capacty. The paper s organzed as follows. In secton 2 we present prelmnary consderatons related to a network wth all the lnks havng an nnte capacty. The case of a nte capacty lnk s consdered n secton 3. The correspondence wth the generalzed Engset system s gven, and the requred blockng probabltes are derved. An example of lnk occupancy and call blockng calculatons s presented n secton 4. Secton 5 gves a bref summary. 2 Lnk occupancy n an nnte system In ths secton we consder a lnk n a system, whch has nnte capacty on all ts lnks, and where, consequently, no blockng occurs. Frst we set the mathematcal model up ncludng the assumptons needed later on. Then we determne the mean tmes that an ndvdual channel traversng the lnk s on and o. Fnally we show how to calculate the dstrbuton for the lnk capacty usage. The results wll be utlzed n the next secton, where we focus on the blockng problem of a lnk wth nte capacty. Consder a lnk n an nnte system. The multcast channel populaton s denoted by I,.e., I s the set of channels (\programs") provded by the orgnatng oce. Let c 2 Z + denote the capacty requrement of channel 2 I. We assume that the users downstream of the consdered lnk subscrbe to these channels accordng to a Posson process wth ntensty. Ths s a model for an nnte user populaton, whch s a reasonable assumpton n networks wth a large number of users, such as TV or rado multcastng n a network (for a lnk not too close to the leaves of the multcast tree). Further, we assume that each user chooses the channel ndependently of others and from the same preference dstrbuton, beng the probablty that channel s chosen. As a result, the subscrptons to channel arrve accordng to a Posson process wth ntensty. We assume that the users' holdng tmes are generally dstrbuted wth mean,1. Fnally, let a denote the oered trac ntensty for channel, a,1 : (1) Consder then the on and o tmes (also called the Page 2

3 duty and dle tmes, respectvely) of a sngle channel. Let (1) ;on and T ;o denote ther means, respectvely. As mentoned above, no blockng occurs n an nnte system. Thus, f the channel s o, t s turned on every tme a new subscrpton arrves. The channel remans n the on state (and occupes the lnk) as long as there are users connected to the channel. Thus, the probablty p that channel s on equals the probablty that there s at least one user connected to the channel. The probablty q that channel s o s then the same as the probablty that there are no users connected to the channel. On the other hand, under the assumptons made above, the number of users smultaneously connected to channel s dstrbuted as the number of customers n an MG1 queue,.e. accordng to a Posson dstrbuton wth mean a. Thus, p 1, e,a ; (2) q e,a : (3) Another mplcaton s that duty and dle tmes of the channel consdered are dstrbuted as busy and dle perods, respectvely, n the correspondng MG1 queue. Thus, ;on ea, 1 ; (4) ;o,1 : (5) The former equaton follows from the fact that p ;on ;on + ;o : (6) We see that the mean duty tme of the most popular channels grows extremely rapdly because of the exponental term n the numerator. Ths ndcates that there s lkely to be a set of channels that are almost constantly carred on the lnk. Let X denote the number of channels n use. Snce X X 2I X ; (7) where X are ndependent Bernoull varables wth mean p,wehave E[X] X 2I p ; (8) Var [X] X 2I p q : (9) Let then Y denote the number of capacty unts smultaneously occuped n the lnk, Y X 2I c X : (10) Its dstrbuton ( j ) 1 j0, called the lnk occupancy dstrbuton, can be calculated by the convoluton algorthm [4], or, equvalently, from the probablty generatng functon: P (z) Y 2I(q + p z c ) 1X j0 j z j : (11) As regards the mean and varance of Y, t follows from (10) and the ndependence of the Bernoull varables X that E[Y ] X 2I Var [Y ] X 2I c p ; (12) c 2 p q : (13) 3 Blockng n a lnk wth nte capacty When multcast connectons are carred on a lnk whch has nte capacty, C, blockng may occur. In ths secton we showhow to calculate blockng n such a lnk assumng that all the other lnks have nnte capacty. It s mportant tomake a dstncton between varous types of blockng. The channel blockng B c of channel s dened to be the probablty that an attempt to turn channel on fals due to lackng capacty, whereas the call blockng b c of channel (seen by a user subscrbng to channel ) refers to the probablty that a user's attempt to subscrbe to channel fals. These are derent, snce the user's subscrpton s always accepted when the channel s already on. Fnally we dene the tme blockng B t of channel to be the probablty that at least C,c +1 capacty unts of the lnk are occuped. Consder a sngle channel 2 I. Denote by T ;on and T ;o the mean on and o perods, respectvely,n ths nte system. By consderng a cycle consstng of an on perod and the followng o perod, we deduce Page 3

4 that the call blockng of channel s b c T ;o, 1 T ;on + T ;o ; (14) where T ;o, 1 s the mean number of faled attempts to subscrbe to channel durng the cycle (the last subscrpton arrvng n the o perod wll be accepted). The frequency of accepted calls when the channel s o s clearly (1, B c ). Thus, 1 T ;o (15) (1, B c ): On the other hand, we observe that n ths nte system (where the capactes of all the other lnks are assumed to be nnte) the on perod of a channel obeys a dstrbuton totally ndependent of the evoluton of the other channels: once the channel s turned on all the ncomng subscrptons wll be accepted. Ths mples that the on perods are dstrbuted as those of an nnte system. Thus, T ;on ;on ea, 1 : (16) By usng equatons (14), (15) and (16), we obtan the followng expresson for the call blockng probabltes of channel : b c B c (1, B c )(ea, 1) + 1 : (17) Thus, the only tem that stll remans to be determned s the channel blockng B c. We start the dervaton by observng that our nte system can be descrbed as a generalzed Engset system wth undentcal users and generally dstrbuted holdng tmes. The channels n our system represent the users n the Engset system. When the channel s on, the \user" s actve, and when the channel s o, the \user" s dle. Thus, the holdng tme of user n the generalzed Engset system s generally dstrbuted wth mean T ;on, and the nterarrval tme s exponentally dstrbuted wth mean,1. As a consequence, we deduce that the channel blockng B c equals the call blockng of user n the correspondng generalzed Engset system. Smlarly, the tme blockng B t equals that of the generalzed Engset system. The tme blockng of user n the generalzed Engset system can be calculated from the followng formula: B t P C jc,c+1 j P C ; (18) j0 j where j s the probablty that j capacty unts are occuped n an nnte system as dened n equaton (11). In the specal case that c 1 for all, ths follows from the result of [1]. In the general case, t can be shown to follow from the nsenstvty property of the product form probabltes of a multrate loss system [9]. It s also known that the call blockng of user equals the tme blockng (of user ) na system where user s removed. Thus, B c P C jc,c+1 () j P C ; (19) j0 () j where () j s the probablty that j capacty unts are occuped n an nnte system wth user removed. These occupancy probabltes can be dented from the probablty generatng functon 1X j0 () j z j Y k2i,fg (q k + p k z c k ); (20) where I,fg denotes the reduced set of users. Alternatvely, n order to save computatonal eort, one may use deconvoluton, whch means Y k2i,fg (q k + p k z c k ) P (z) ; (21) q + p zc but numercal problems may arse for large systems. To summarze, the call blockng b c can be calculated from formula (17) by usng (19). Note that the denomnator n (17) s always greater than 1. Thus, the call blockng b c seen by a user subscrbng to channel s always smaller than the correspondng channel blockng B c. Ths reects the fact that the users subscrbng to a channel whle the channel s on do not experence any blockng. We see also that, for the most popular channels, blockng seen by a user drops practcally to zero, snce the exponental term n the denomnator grows rapdly wth a (b c B c e,a ). For a channel wth a 1, the channel blockng and the call blockng seen by a user are approxmately the same. Snce b c B c B t, an upper lmt for the call blockng s the tme blockng n a system wth all channels present. No call blockng seen by a user can be hgher than ths, but call blockng for user approaches t for channels wth channel preferences near zero. Page 4

5 4 An Example As an example, a truncated geometrc dstrbuton s used for channel choosng preferences: / (1, p),1 ; 1; 2;:::;jIj; (22) where the ndces are the channel numbers arranged n a descendng order accordng to ther usage and a value of p has been chosen to be p 0:2. We have chosen the average vewng tme,1 900 s to be the same for all channels, and,1 3: The lnk capacty s C 70 channels, and each channel has dentcal capacty requrement c 1. We have calculated blockng wth jij 512 channels, but wll present only the rst 200 channels. The rest may be extrapolated from the graphs. The channel usages p are depcted n gure 3 as a functon of channel ndex Channel usages 10 0 Lnk occupancy Number of channels Fgure 4: Lnk occupancy dstrbuton for geometrcally dstrbuted user preferences Channel on tme Channel usage sec Channel ndex Fgure 3: Channel usage versus channel ndex for geometrcally dstrbuted user preferences Channel ndex Fgure 5: The mean duty tmes T ;on n seconds for each channel. The mean and varance of the number of used channels are calculated from equatons (8) and (9). They are 63.4 and 3.11, respectvely. In ths example, where all capacty requrements are equal to 1, these same values hold for mean and varance of capacty requrement for the channels. The lnk occupancy dstrbuton ( j ) s presented n gure 4. The mean channel duty tmes T ;on for ths example are presented n gure 5. As expected, the most popular channels have practcally nnte duty tmes. Call and channel blockngs for each channels are shown n gure 6. 5 Summary In ths paper, we have presented a method for calculatng the call and channel blockng probabltes n a lnk carryng multcast trac. Multcast trac has the property of requrng less lnk capacty than a set of pont-to-pont connectons provdng the same connectvty. The blockng calculaton presented gves us a grp of TV or rado delvery on a crcut swtched system, such as an ATM network wth vrtual crcuts. We are able to calculate the capacty needed for a lnk, the average channel duty tmes, and blockng ntroduced by nte capacty. Page 5

6 Blockng 10 4 B & b Channel ndex Fgure 6: Call B c and channel blockngs b c for each channel (B c sold lne, dotted lne). bc The calculaton started from a dstrbuton of the users' preferences on multcast channels, from whch we derved the lnk occupancy dstrbuton n an nnte capacty system, and, further, the blockng probabltes n a nte system by mappng the problem to an equvalent generalzed Engset system. Fnally, an example was gven wth the geometrc dstrbuton of users' preferences. In ths paper, we have lmted ourselves to the case of a sngle lnk. The case where blockng occurs on several lnks of a network s left for future work. Possbly methods such as Reduced Load Approxmaton [9] can be used, but blockngs on successve lnks may exhbt sgncant dependences. It s also lkely that the users' actual preference dstrbuton aects sgncantly the blockng ntroduced n the network. Ths part of the study would requre statstcs from a real-lfe trace, and s left for future work. Acknowledgements The authors wsh to thank prof. Gunnar Karlsson, Petter Koponen and Juhana Rasanen from the Helsnk Unversty of Technology for ther nsprng thoughts. (November 1957), 158{170. [2] Dot, C., Dabbous, W., and Crowcroft, J. Multpont communcaton: A survey of protocols, functons, and mechansms. IEEE Journal on Selected Areas n Communcatons 15, 3 (Aprl 1997), 277{290. [3] Gacomazz, P., and Trecord, V. A study of non blockng multcast swtchng networks. IEEE Transactons on Communcatons 43, 2/3/4 (March 1995), 1163{1168. [4] Iversen, V. B. The exact evaluaton of mult-servce loss systems wth access control. Teleteknk, Englsh Edton 31, 2 (August 1987), 56{61. [5] Kaufman, J. S. Blockng n a shared resource envronment. IEEE Transactons on Communcatons com-29, 10 (October 1981), 1474{1481. [6] Km, C.-K. Blockng probablty of heterogenous trac n a multrate multcast swtch. IEEE Journal on Selected Areas n Communcatons 14, 2(February 1996), 374{385. [7] Naoumov, V. Normal-type approxmaton for mult-servce systems wth trunk reservaton. Telecommuncaton Systems, 4 (1995), 113{118. [8] Roberts, J. W. A servce system wth heterogenous user requrements { applcaton to mult-servces telecommuncatons systems. In Performance of data communcaton systems and ther applcatons (Amsterdam, September 1981), G. Pujolle, Ed., North-Holland, pp. 423{ 431. [9] Ross, K. W. Multservce Loss Models for Broadband Telecommuncaton Networks. Sprnger Verlag, London, [10] Shacham, N., and Yokota, H. Admsson control algorthms for multcast sessons wth multple streams. IEEE Journal on Selected Areas n Communcatons 15, 3 (Aprl 1997), 557{ 566. References [1] Cohen, J. W. The generalzed Engset formulae. Phlps Telecommuncaton Revew 18, 4 Page 6