Analysis of Queuing Delay in Multimedia Gateway Call Routing

Transcription

1 Analyss of Queung Delay n Multmeda ateway Call Routng Qwe Huang UTtarcom Inc, 33 Wood Ave. outh Iseln, NJ 08830, U..A Errol Lloyd Computer Informaton cences Department, Unv. of Delaware, Newark, DE 976, U..A. Abstract: The recent emergence of multmeda applcatons has led to the deployment of varous telecommuncaton technologes (e.g. IP, wreless) n addton to tradtonal PTN (Publc wtched Telephone Network) crcut swtched networks. In ths heterogeneous telecommuncaton world, users make calls wthout regard to the fact that the orgnatng network may dffer from the termnatng network, the call may be routed through one or more ntermedate networks. For nstance, users from two dfferent PTN networks may communcate through an IP network usng Voce over IP (VoIP) technology. In ths framework, multmeda gateways perform call routng, call sgnalng conversons meda format conversons among dfferent networks. Ths paper nvestgates algorthms queung delays on multmeda gateways n regard to call routng amed at mnmzng the expected call delay. Both centralzed dstrbuted queung models are consdered, performance measures are derved. eywords: multmeda gateway, VoIP, call routng, queung delay, optmzaton. Introducton In telecommuncatons, multmeda gateways have recently emerged as a tool for provdng realtme, mult-way communcaton among dfferent meda (e.g. IP, PTN, wreless etc). The functonaltes of multmeda gateways nclude call routng, call sgnalng conversons meda conversons []. Among these functonaltes, ths paper focuses on expected queung delays assocated wth call routng. In the remer of ths secton, we brefly descrbe multmeda gateway archtectures functonaltes. In secton we descrbe the problem that we consder. In secton 3 we present our algorthms analyss.. eneral Background Fgure Multmeda gateway nterconnectons. A typcal multmeda gateway nterconnecton s shown n Fgure. There, two multmeda gateways, NYC LA, are nterconnected by three meda networks: IP, PTN wreless. Each end user s connected to a sngle local network. Each local network s connected to a sngle gateway. Multmeda gateways perform three hgh-level functonaltes: call sgnalng conversons, meda conversons call routng. pecfcally, call sgnalng conversons are to convert sgnalng messages n one type of network ( such as VoIP call sgnalng H.5 H.45) to the sgnalng messages n another type of network (such as PTN call sgnalng Q.93). mlarly, meda conversons are to convert the meda format provded n one type of network (say, PTN D0) to the meda format requred n another type of network (such as VoIP RTP packets wth dfferent codecs). nce call sgnalng conversons meda conversons are not relevant to ths paper, nterested readers can refer to [] [] for addtonal detals. Ths paper studes call routng as detaled below.. Call Routng When a call arrves at a multmeda gateway from a local meda network, that gateway must allocate that call to one of the non-local meda networks assocated wth the gateway, ths s the functonalty of call routng. Here, a channel s a physcal resource that transmts receves voce/data nformaton. Each call s processed by one channel, the nature of whch s network specfc: In an IP network, a channel refers to a DP (dgtal sgnal processor) channel; In a PTN network, a channel refers to a D0 channel; In a

2 wreless network, a channel refers to a rado channel. The (gateway) bwdth s the total number of calls that can be processed smultaneously s equal to the total number of avalable channels taken over all meda. Each meda s bwdth s the number of channels avalable on that meda. The number of smultaneous calls on each meda can t exceed ts bwdth. On multmeda gateways, channels on each meda are usually dvded statcally nto two groups. One group s used for ncomng calls the other group s used for outgong calls. We assume that there s no delay to process ncomng calls from other gateways local networks: ncomng calls are ether accepted or rejected nstantaneously dependng on channels avalablty. From ths assumpton, the queung delay of a multmeda gateway s ndependent of the queung delay on the other gateways. Thus, throughout ths paper, our focus s on a sngle multmeda gateway s call routng n ts non-local meda networks..3 Centralzed Queung Model vs Dstrbuted Queung Model Fgure llustrates the nternal archtecture of a multmeda gateway. A central controller s connected to each meda controller through a control bus. Each meda controller has ts physcal meda nterface connected to the meda network, t s responsble for the meda access control. The central controller s responsble for the operaton, admnstraton management of the entre system, t s also responsble for the call routng. Queung delay occurs when a call can t be processed mmedately upon ts arrval. There are two types of queung models: One model s centralzed queung: the queung of the calls s centralzed at the central controller. The central controller decdes when to route process a call on whch meda dependng on real tme nformaton about channel utlzaton as provded by each meda controller. pecfcally, each meda controller nforms the central controller (through the control bus) as soon as a call on that meda s completed. The other queung model s dstrbuted queung: here, the queung of the calls s dstrbuted nto each meda controller. Each meda controller decdes when to process the calls routed to t (by the central controller) based on ts own channel utlzaton. In secton 3., we wll present two routng algorthms for these two queung models respectvely. Fgure Multmeda gateway nternal archtecture. Problem Descrptons As noted above, the call routng functon of a multmeda gateway s to decde whch meda wll process each call. Relatve to call processng, n ths paper we make the followng stard assumptons: each call should be processed by exactly one channel cannot be pre-empted; one channel can process only one call at a tme; once a call s routed to a meda, the call s processed mmedately f there s an dle channel, otherwse, the call wll be placed nto the watng queue s processed later when a channel becomes dle on a frst come, frst served bass. Consderng a sngle multmeda gateway, the goal of routng s to mnmze the expected queung delay of calls. Assocated wth ths problem, there are two sets of system parameters. One set of parameters s assocated wth calls: the call arrval rate s a Posson process wth parameter each call has an exponental call length wth parameter µ. We assume that the buffer sze of the queue (for calls that cannot be processed mmedately) s nfnte. The other set of parameters s assocated wth meda: there are non-local meda networks connected to the multmeda gateway; the bwdth of M s B, whch s the maxmum number of calls that can be processed by M at any gven tme of nstance. Let B = B be the total bwdth of all meda. Relatve to centralzed queung model dstrbuted queung model, a centralzed queung gateway has one central queue n the central controller (Fgure 3). The central controller routes a call mmedately f there s an dle channel on any of ts meda; otherwse, t places the call nto the =

3 central queue routes t later on a frst come frst served bass as soon as a channel becomes avalable on any of ts meda. A dstrbuted queung gateway has one queue n each of ts meda controllers (Fgure 4). After the central controller routes a call to one of the meda controllers, the meda controller processes the call mmedately f there s an dle channel on that meda; otherwse t places the call nto ts queue processes t later on a frst come frst served bass as soon as a channel on that meda becomes avalable. Fgure 3 Centralzed queung model. Fgure 4 Dstrbuted queung model. 3 Routng Performance Results In ths secton, we present our man results. In secton 3., we outlne several useful results wth respect to M/M/m queues. In sectons , we descrbe two routng algorthms: a greedy algorthm a traffc splttng algorthm. These two algorthms are desgned respectvely for the centralzed queung model the dstrbuted queung model. ectons also contan analyss of the algorthm performance, for traffc splttng, a determnaton of optmal parameters for use n that algorthm. In secton 3.4, we compare the performance of the two algorthms. 3. Propertes of an M/M/m Queue In ths secton, we provde (wthout proof, due to space) some queung propertes for M/M/m queues. Recall that the call arrval rate s a Posson process wth parameter that each call has an exponental call length wth the parameter µ. Thus, an M/M/m queue [3] models a multmeda gateway that conssts of a sngle queue can smultaneously process a maxmum of mm ( ) calls. Let P be the probablty that there are calls n the system. The expected watng tme W of a call n the queue s gven by [3]: m m m + ρ W = P0 (.) m!( ρ) Where ρ = /mµ < (.) m m n m m ρ ( mρ) 0 = + m!( ρ) n= 0 n! P (.3) uppose that m µ are fxed ρ s a varable. From (.), s also a varable, W s a functon of ρ (or ) for the gven m µ. The relatonshp between W ρ s as follows: dw ( ρ) d W ( ρ) Theorem (.4) > 0, > 0 ( ρ < ). dρ dρ dw ( ) d W ( ) Corollary (.5) > 0, > 0 ( <m µ ). d d Theorem (.4) corollary (.5) show that for fxed m µ, W dw / d ρ are monotone ncreasng functons wth respect to ρ ( ρ <) ( <m µ ). Now suppose that ρ µ are fxed, thus / m s fxed. From (.) (.), m m ρ W = P m 0. Thus W s a functon of m. µ m!( ρ) Theorem (.6) If ρ ( ρ <) µ are fxed, then W( m ) s a monotone decreasng functon. Let D( m) = W( m+ ) W( m) be the changng rate of W( m ), then D( m ) s monotone decreasng. Theorem (.6) shows that for the fxed m µ, the queung delay W( m ) the changng rate D( m) of W( m ) decrease when m ncrease proportonally (see also Example II n secton 3.3.).

4 3. A reedy Algorthm for Centralzed Queung Model In ths secton we descrbe a smple algorthm for centralzed queung model, derve the expected queung delay for that algorthm, prove that the algorthm mnmzes the expected queung delay among all algorthms. 3.. The reedy Algorthm A natural approach to centralzed queung model s to utlze a tradtonal greedy algorthm. Here, upon the arrval of a call, f there s an dle channel on some meda, the algorthm routes the call to that meda where the call can be processed mmedately; otherwse, the call s placed nto the central queue s routed later on a frst come frst served bass when a channel becomes dle. pecfcally: Algorthm (.) reedy_routng() { whle () f (there s an ncomng call) f (the central watng queue s not empty) put the call nto the central queue; else f there s an dle channel on any meda Routng (the call); else put the call nto the central watng queue; whle (there are dle channels due to call completon the watng queue s not empty) get one call from the central watng queue; Routng (the call); } Routng (the call) { for = to do f there s an dle channel n } route the call to M 3.. Performance Analyss Recall that = M B = B s the total bwdth of all meda. If we let m=b, from the greedy algorthm secton 3., we know that the expected queung tme of a call s the same as the expected queung tme for an M/M/m queue. Thus, we have the followng theorem: Theorem (.) Let W be the expected queung tme of a call generated by reedy_routng. Then m m m ρ W = P 0 wth ρ = < µ m!( ρ) mµ m m m n m ρ ( mρ) P0 = +. m!( ρ) n= 0 n! Theorem (.) shows that the expected queung tme of a call W depends only on, µ the total bwdth m=b, but not on each ndvdual meda s bwdth: Example I: =, m =, m =, =, µ =. ρ= = = <. Thus, W = 4/9. m µ (m + m) µ 3 Although the greedy algorthm seems apparent, t s n fact optmal as the next result shows: Theorem (.3) Among all algorthms for centralzed queung gateway, the greedy algorthm mnmzes the expected queung delay. (The proof s omtted due to space). 3.3 The Traffc plttng Algorthm for Dstrbuted Queung Model A traffc splttng algorthm s used n dstrbuted queung gateways, whch can be modeled as a combnaton of M / M / B ( =,,..., ) queues. The central controller routes the calls (splts the traffc) to each meda controller wth predetermned probablty to acheve the expected mnmum delay. In ths secton, we frst show how to derve the predetermned probablty to acheve the expected mnmum prove ts correctness. Then we descrbe the traffc splttng algorthm. Fnally we present some examples Calculaton of Predetermned Probabltes The analyss below focuses on how to obtan the predetermned probabltes to mnmze the expected delay from the dstrbuton of call arrval rate call length as well as the bwdth of each meda. Let W( =,,... ) be the expected watng tme for a call routed to the meda the expected watng tme of a call: W = pw = W = = M, W be (3.)

5 It s obvous that p = (3.) = It follows that = p (3.3) = (3.4) = We need to fnd the probablty vector ( p, p,... p ), or equvalently (,,... ), to mnmze W. Let meda M s bwdth B = m m = m. uppose that we have found such a = vector. nce the arrval rate of a call at meda M s a Posson process wth rate = p [3], W can be decded accordng to (.), (.) (.3): m m+ m ρ W = m m m n!( ) (3.5) m ρ ( mρ) m ρ + m!( ρ ) n! n= 0 where ρ = /m µ < (3.6) From (3.4) (3.6), < µ m (3.7) = If =, W s fxed. Thus we assume >. Let W f f = s = ( =,,... ). Accordng to the corollary (.5), we have W W f = = ( W + ) > 0 (3.8) f W W ( s = = + ) > 0 (3.9) W We let =0 ( j =,,... ) j (3.0) From (3.4), we treat ( =,,... ) as ndependent varables, depends on ( =,,... ). Thus = (3.) = (for < ) We smplfy (3.0) usng (3.), = (3.) W W W = + + f j j j j, = j = f f =0 ( j =,,... ) (3.3) j We have the followng theorem corollary: Theorem (3.4) There exsts a unque (,,... ) to (3.5), (3.6), (3.7), (3.) (3.3) that mnmzes W. (The proof s omtted due to space) Corollary (3.5) When m = m =... = m, the expected watng tme W s mnmzed ff =... = = /. enerally, the exact soluton of the vector (,,... ) s unattanable snce (3.3) are nonlnear equatons. But an approxmate value of the vector (,,... ) can be obtaned by usng Newton s method for the non-lnear equatons descrbed n [4], we wll not dscuss t here The Algorthm From theorem (3.4), we descrbe the traffc splttng algorthm as follows: Algorthm (3.6) Traffc_plttng() { calculate (,,... ) from (3.5), (3.6), (3.) (3.3) by usng Newton s method for the non-lnear equatons descrbed n [4]. for each call, do route the call to M wth probablty / } Next, we gve two examples to calculate the probablty to acheve the expected mnmum delay: Example II: =, m =, m =, =, µ =. /(m + m) µ = /3<. From (3.5), we have W =, W = ( ) 4. From (3.), we have 3 W = W + W = + ( ) (4 ) (3.7) From (3.), we have = (3.8) 3 ( ) Thus, W = + ( ) (4 ) (3.9) Let dw 3 / d = 0, = 0 (3.0)

6 Usng Newton s method, we have =.4, = 0.589, p = , p = 0.945, W =.4. Note that although the second meda has double the capacty of the frst meda, more than two thrds of the calls are routed to that meda n an optmal soluton. The reason s as follows: f we route exactly two thrds of the calls to the second meda, then ρ = ρ = /3. From theorem (.6), we know that the changng rate of the expected queung delay n the frst meda s larger than that n the second meda, whch means that f we route more calls to the second meda, we can further decrease the expected queung delay. Example III: =, m = 8, m =8, = 8, µ =. Immedately from corollary (3.5), we know that when = = /= 4, (, ) mnmzes the expected delay for a call. Thus p = p = Performance Comparson: Centralzed Queung vs Dstrbuted Queung In ths secton, we compare the expected queung delay of the greedy algorthm (for centralzed queung model) the traffc splttng algorthm (for dstrbuted queung model), as descrbed n the pror two sectons. Note that the queung delays that arse n centralzed queung gateway (modeled by an M/M/m queue where m = B ) versus dstrbuted = queung gateway (modeled by M / M / B ( =,,..., ) queues) result from slghtly dfferent sources. In centralzed queung model, the delay s due to the postponement of routng on the central controller untl a channel becomes avalable on some meda. There, no queung delay results from each meda controller. In contrast, n dstrbuted queung model, there s no delay due to routng on the central controller. Rather, queung delays occur as a result of watng to process the call untl a channel becomes avalable on a partcular meda. As one mght expect, we have the followng theorem, whch establshes that centralzed queung model can never be worse than dstrbuted queung model. The proof s omtted due to space. Theorem (3.) Let (>) be the number of meda let meda M s bwdth B = m m= m. If / mµ <, then W < W. = Examples I II (shown earler) are examples of applyng the greedy traffc splttng algorthms for the same parameters. It can be easly seen that W < W n these two examples. The followng theorem (the proof s omtted due to space) shows that when each meda has the same bwdth, W s at least tmes W : Theorem (3.) ven (>) meda, where each meda has same bwdth B, let m = B = B = W W = mnw mn. Then f / mµ <, < ; W f / mµ, W / Wmn /. Example IV: =, m = m =, = 3.6, µ =. 3.6 ρ = 0.9. W 4.63, mn (m + m ) µ = 4 = < = W =.0. Thus r = W / W = < / = / mn Example V: =, m = m =, = 3.96, µ = ρ = 0.99, W 49.5, mn (m + m ) µ = 4 = < = W = Thus r = W / W = < / = / mn From these two examples, we can see that r s closer to / than r snce ρ s closer to than ρ. nce centralzed queung model can never be worse than dstrbuted queung model, then what are the advantages of dstrbuted queung model? By usng dstrbuted queung model, the central controller doesn t need to know the real tme channel usage nformaton from each meda controller, the queung functonalty s located nsde each meda controller. Thus the workload of the central controller can be allevated. References [] ITU-T H.33 Packet-Based Multmeda Communcatons ystems. eres H: /000. [] ITU-T H.48 Meda ateway Control Protocol, eres H: A.M..., 06/000. [3] R. Nelson, Probablty, tochastc Procedures, Queung Theory. prnger-verlag, 000. [4] L. V.Fausett, Appled Numercal Analyss Usng MetLab. Prentce Hall, 999.

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