Quality-of-Service Routing in Heterogeneous Networks with Optimal Buffer and Bandwidth Allocation

Transcription

1 Purdue Unversty Purdue e-pubs ECE Technca Reorts Eectrca and Comuter Engneerng Quaty-of-Servce Routng n Heterogeneous Networs wth Otma Buffer and Bandwdth Aocaton Waseem Sheh Purdue Unversty, waseem@urdue.edu Arf Ghafoor Purdue Unversty, ghafoor@urdue.edu Foow ths and addtona wors at: htt://docs.b.urdue.edu/ecetr Sheh, Waseem and Ghafoor, Arf, "Quaty-of-Servce Routng n Heterogeneous Networs wth Otma Buffer and Bandwdth Aocaton" (007. ECE Technca Reorts. Paer 45. htt://docs.b.urdue.edu/ecetr/45 Ths document has been made avaabe through Purdue e-pubs, a servce of the Purdue Unversty Lbrares. Pease contact eubs@urdue.edu for addtona nformaton.

2 Quaty-of-Servce Routng n Heterogeneous Networs wth Otma Buffer and Bandwdth Aocaton Waseem Sheh and Arf Ghafoor Schoo of Eectrca and Comuter Engneerng, Purdue Unversty West Lafayette, IN Ema: {waseem, ghafoor}@ecn.urdue.edu Abstract We resent an nterdoman routng rotoco for heterogeneous networs emoyng dfferent queung servce dscnes. Our routng rotoco fnds otma nterdoman aths wth maxmum reabty whe satsfyng the end-to-end tter and bandwdth constrants n networs emoyng heterogeneous queung servce dscnes. The quaty-of-servce (QoS metrcs are reresented as functons of n bandwdth, node buffers and the queung servce dscnes emoyed n the routers aong the ath. Our scheme aows smart tunng of buffer-sace and bandwdth durng the routng rocess to adust the QoS of the nterdoman ath. We formuate and sove the bandwdth and buffer aocaton robem for a ath over heterogeneous networs consstng of dfferent queung servces dscnes such as generazed rocessor sharng (GPS, acet by acet generazed rocessor sharng (PGPS and sef-coced far queung (SCFQ. Keywords Quaty-of-servce (QoS, resource aocaton, nter-doman routng, queung servce dscnes. I. INTRODUCTION Wth the evouton of the Internet nto a goba communcatons medum consstng of heterogeneous networs, the need for rovdng quaty-of-servce (QoS guaranteed mutmeda data over heterogeneous networs has ncreased tremendousy. To rovde end-to-end QoS guarantees, the nterdoman ath routng rotoco must tae nto account the resources such as bandwdth and buffer, and heterogenety of the networs comrsng the Internet. In ths aer we consder the heterogenety arsng from the dfferent queung servce dscnes used n the routers n a doman. Ltte wor has been done to ncororate the nformaton about resources and queung servce dscnes nto QoS routng. Present nterdoman routng rotocos such as border gateway rotoco (BGP do not rovde such mechansm []. We resent an nterdoman routng rotoco whch reresents QoS metrcs as functons of buffers, bandwdth and queung servce dscnes rather than statc metrcs. Our rotoco uses ths functona reresentaton of QoS metrcs to fnd otma nterdoman aths wth maxmum reabty whe satsfyng end-to-end tter and bandwdth constrants. Our routng rotoco uses the nowedge of dfferent queung servce dscnes n mute domans to fnd the otma nterdoman ath. Our nterdoman routng rotoco has the foowng advantages over exstng nterdoman routng rotocos: Frst, our routng rotoco s aware of the reatonsh among QoS metrcs and resources beongng to heterogeneous networs and fnds an otma nterdoman ath by eeng ths reatonsh nto consderaton. Second, t modes the QoS metrcs as functons of resources such as buffer and bandwdth. Thrd, t catures the nterdeendency between varous QoS metrcs such as reabty and tter deay. Ths aer s organzed as foows. In the next secton we rovde an overvew of the reated wor n the area of QoS routng. In Secton III, we formuate and rovde an otma souton to the nterdoman ath resource aocaton robem and resent an agorthm for nterdoman QoS routng wth resource aocaton robem. Ths agorthm uses the souton to the nterdoman ath resource aocaton robem as a subroutne. Fnay, n Secton IV we concude the aer. II. RELATED WORK The research n the area of QoS routng has many focused on two aroaches: QoS routng wthout resource aocaton [], [] and QoS routng wth resource aocaton [4], [5], [6], [7], [8]. In QoS routng wthout resource aocaton the QoS metrcs are modeed as non-negatve ntegers. In determnng these metrcs the nteracton among resources s gnored. In addton the metrcs aong a ath are smy added [], []. The QoS routng robem formuated n ths manner s a constraned shortest ath robem.

3 In QoS routng wth resource aocaton there are two aroaches. In [4], [5], [6] resources are defned as numerca vaues assocated wth each edge of the grah. The QoS metrcs assocated wth each edge are comuted by usng the avaabe resources, traffc descrton and the queung servce dscne. In ths aroach the QoS metrcs and resources are statc and can not be fne-tuned durng the routng rocess. In [7], [8] the QoS metrcs are reresented as functons of the bandwdth, buffer and the queung servce dscne. Ths defnton ncororates the reatonsh between QoS metrcs and resources. However the authors do not consder nterdoman routng robem over heterogeneous networs whch catures the reatonsh between dfferent queung servce dscnes. In ths aer we formuate and sove ths robem. III. PROBLEM FORMULATION AND SOLUTION The networ consstng of mute domans (autonomous systems s modeed by a drected grah G ( V, E =, where V s the set of nodes and E s the set of edges n the grah. In each AS (autonomous system a artcuar queung servce dscne s memented n the outut queue of every router beongng to that AS. Mute AS' s may have dfferent queung servce dscnes memented n them. As a seca case of the above networ, we consder a grah G consstng of three AS' s as shown n Fg.. The queung servce dscnes that are used n the three AS' s are generazed rocessor sharng (GPS [9], acet by acet generazed rocessor sharng (PGPS [0] and sef-coced far queung (SCFQ []. We formuate and sove the nterdoman QoS routng wth resource aocaton robem for the seca case of the three AS' s shown n Fg.. Ths resut can be generazed to more than three AS' s emoyng varous queung servce dscnes. Mutmeda Server ( m Cent Inter-doman router ( n m ( Autonomous System SCFQ Autonomous System GPS Autonomous System PGPS Fg.. An nterdoman ath over heterogeneous networs emoyng GPS, PGPS and SCFQ queung servce dscnes. Let 0, n = v, v, v..., v, v, v, v,..., v, v, v, v,..., v be a ath n the grah G n Fg., where v m m m m n G, mn,, are ostve ntegers such that > 0 and < m< n. The ath ( n = ( ( m ( n m where ( = v0, v, v,..., v, v, ( m v, v+, v+,..., vm, vm and ( n m vm, vm+, vm+,..., vn and the oerator actng on ( n and ( m as ( n ( m concatenates ( n and that the ast node of ( n and the frst node of ( m are common. In our anayss we consder that ( ( m AS and ( n m beongs to n can be wrtten as = = m such beongs to AS, beongs to AS as shown n Fg.. We consder that the queung servce dscnes memented n AS, AS and AS are GPS, PGPS and SCFQ resectvey. Notce that the node v s common to AS and AS and node v m s common to AS and AS. The nodes v and v m reresent the nterdoman routers. Wthout oss of generaty we assume that PGPS s memented n the outut queue of v and SCFQ s memented n the outut queue of v m. ( Let the data traffc arrva functon at the outut queue of node v be A out ( tv,. We use Leay Bucet constraned sources to ( mode the data sources. Aout ( tv, s sad to conform to (, ρ f for any nterva ( τ,t], Aout ( t, v Aout ( τ, u + ρ ( t τ. In our anayss, we assume that the traffc enterng AS conforms to (, ρ, the traffc enterng AS conforms to (, ρ and the traffc enterng AS conforms to (, ρ. The traffc shang mechansm s memented n the nterdoman routers, n our case n the nodes v 0, v and v m. We defne the nterdoman ath resource aocaton robem for the ath ( n as foows:

4 ( = buv (,, ruv (, ; ( uv, n { ( } Q n max Q { } n P subect to: J n J C ( b ( u v B( u v ( u v ( n ( C 0,,,, max (, (,, (, ( ( R r u v C u v u v n C The symbos used n ( are defned n TABLE I. Q ( ( n s the maxmum reabty of the ath ( varabes n the otmzaton robem ( are the n buffer sze ( b ( u, v and the n bandwdth r ( u, v for a ns ( uv, beongng to the ath ( n. ( C moses the constrant that the tter for ath maxmum aowabe tter vaue, J. Constrant ( avaabe. ( C moses the constrant that the n bandwdth aocated, ( r (, mnmum bandwdth uested, R, and can not be greater than the maxmum ossbe n bandwdth, C( u, v max ( the nequaty R r ( u, v resuts from the fact that r ( ( n = mn { r ( u, v }. ( uv, ( uv, n { } We defne the nterdoman QoS routng wth resource aocaton robem as foows: n. The decson n shoud be ess than or equa to the C moses the uer bound on the maxmum hysca buffer sze, = max { Psd (, } Q s d Q u v, shoud be greater than or equa to the where P( s, d s the set of a aths from node s to d. Secfcay for the nterdoman ath ( n descrbed above and shown n Fg., the otmzaton robem. Note that can be reformuated usng TABLE II, TABLE III and TABLE IV. The vaues of arameters n TABLE II, TABLE III and TABLE IV were derved n [7]. Foowng s the reformuated otmzaton robem: ( ( n = { {( Q ( ( Q ( ( m Q ( ( nm }} Q max mn, (, (, { b ( } mn, = max mn mn, b v v mn mn, + m + ( L b v v mn mn, m+ n, + ( m L ( P subect to:

5 4 ( = ( ( ( ( ( ( + + J n J J m J n m mn b (,, = q ( ( m = + + r ( ( ( ( r m ( ( ( ( ( ( C( v, v q n m n K v, v L + J r = m+ n m 0 b v, v B v, v,,,..., n C ( C { } ( max ρ ρ ρ r ( ( n C( ( C ( max,, mn, n ( where ( ( q n s defned accordng to the foowng recursve reatonsh [7]: { } { } ( ( ( ( ( ( ( ( ( ( 0 + ( 0, = mn ( 0,, + mn,, q = q + b v v + L q b v v b v v L v through n where the ath ( s obtaned by addng the vertex v, to the ath ( bandwdth of a ath s the bottenec functon of the bandwdth of the subaths, therefore we can reace ( ( ( ( ( r ( ( m and r ( ( n m n ( by r ( ( n. A. Souton to the Interdoman Path Resource Aocaton Probem In ths sub-secton we sove the nterdoman ath resource aocaton robem, defned n. Note that snce the r,, and obtan the otma souton. The otma souton conssts of the vaues of the resource varabes.e., the n buffer szes b v, v for a ns (, beongng to the ath r ( ( n and and buffer by n and the ath bandwdth (, b v v resectvey. ( ( ( r n. We denote the otma souton vaues of bandwdth ( Lemma III.: The otma souton vaue for r ( ( n n the otmzaton robem ( s gven by mn C( v max,. n Proof: Let r ( ( n and ( b ( v, be the otma souton vaues for the otmzaton robem (. Let the corresondng otma obectve functon vaue be Q ( ( n. Suose ( ( r ( ( n < r ( ( n where r ( ( n s a feasbe souton of (. Snce r ( ( n and r ( ( n are both feasbe soutons, therefore we can reace r ( ( n by r ( ( n n constrant ( C n otmzaton robem ( and ncrease the vaue of the buffers ( b ( v, aocated ( aong the ath ( n such that ( b ( v, > b (, whe eeng constrant ( C feasbe. Ths mes that we can get another feasbe souton r ( ( n and ( (, b v, and corresondng obectve functon vaue Q ( ( n for ( ( ( such that Q ( ( n > Q ( ( n. Ths eads to a contradcton, snce Q ( ( n was the otma obectve functon vaue. Snce the bandwdth s the bottenec functon of a the ns aong the ath, therefore, t s not ossbe to have the foowng ( nequaty r n > mn C v, v. Hence the otma souton vaue for r n s mn C( v, v max. ( max n ( n

6 Once the otma souton vaue for ( ( sace.e., ( (, b v v aocated aong the ath we can rewrte constrant ( C as beow: r n s determned, we are eft wth determnng the otma vaue for the buffer n. For convenence we defne ( ( C( v, v = m+ 5 n K v, v L J = J. Hence mn b (,, = q ( ( m q ( ( nm + + r ( ( n r ( ( n r ( ( n J ( or equvaenty, mn b (,, + q ( ( m + q ( ( nm J r n = ( ( 4 Lemma III.: If the foowng nequaty hods: ( ( + mn,, B v = m mn B (,, + ( m L+ =+ n mn B (,, + ( nm L J r n = m+ ( ( 5 then an otma souton vaue for ( (, b v v s gven as foows (Note the otma souton may not be unque n ths case: b (, = { ( } { ( } { ( } mn B v, v,, =,,..., mn B v, v, + L, = +, +,..., m mn B v, v, + m L, = m+, +,..., n ( 6 Proof: The nequaty ( 5 states that the maxmum amount of bts that can be buffered n the nodes aong the ath ( n s ess than or equa to the buffer sace ured to voate the maxmum tter vaue J. In other words we can buffer the maxmum number of bts n the nodes aong the ath ( n wthout voatng the tter constrant ( vaue ( souton for b v, v s gven by C. The otma obectve functon Q n can be obtaned by usng the maxmum vaue of the buffer sace avaabe n each node. Hence an otma ( 6. The corresondng otma obectve functon vaue s gven by the foowng:

7 6 Q ( { } mn b v, v b v, v b v, v mn mn mn, ( ( n = + m m+ n + ( L + ( m L ( ( Theorem III.: For the case when the foowng nequaty hods: mn B (,, + = m mn B (,, + ( m L + > J r n =+ n mn B (,, + ( nm L = m+ ( ( 7 then the otma souton ( ( r ( n, b (, to satsfes the foowng: ( max = mn, n r n C v v ( 8 { ( } mn b v, v b v, b, v = mn mn m = m n ( L + ( m L ( ( Proof: ( 8 foows from Lemma III.. The nequaty ( 7 means that the number of bts that can be buffered n the nodes aong the ath ( n can ead to a tter vaue whch can voate the tter constrant. Note that n ths case at the otma souton b v, the nequaty ( 4 w be satsfed as an equaty,.e., ( 9 mn b (,, + q ( ( m + q ( ( n m = J r n = ( ( 0 If ( 4 s a strct nequaty..e., mn b (,, + q ( ( m + q ( ( n m < J r n = ( ( then there exsts ( b ( v, feasbe souton. for at east one {,,..., n} such that Let the obectve functon vaue corresondng to ( (, > (, and ( b ( v, ( (, b be ( b v v b v v s a Q n, then ( ( Q ( ( n > Q ( ( n, whch s a contradcton. Hence ( 0 must hod at otmaty. Now we show that ( 9 must hod at otmaty. Suose by way of contradcton that ( 9 does not hod. Wthout oss of generaty we can assume that the ath ( n = ( ( m. The resut can be extended to a ath consstng of more than two subaths by nducton.

8 7 Assume that at otmaty the foowng hods: { ( } mn b v, v b v, v < mn m + + ( L ( ( Q ( { } mn b v, v b v, v = ( ( n mn + m + ( L ( where Q ( ( n s the otma obectve functon vaue. From t mes that there exsts ε > 0, ε R such that: { ( } mn b v, v, b v v + ε = mn + ( m + ( L ( It s ossbe to reaocate the buffers aong the ath ( n such that ( 0 hods and the new varabes foowng obectve functon: ( ( Q ( { } mn b v, v b v, v ε = + ε = Q + 4 ε mn + m + ( L ( ( (, b v v ead to the ( (.e., Q n > Q n whch eads to a contradcton snce we started of by assumng that Q n s the otma obectve functon vaue. To fnd the otma vaues of buffers ( (, b, we roceed as foows. Let J ( (, J ( ( m ( ( be the tter vaues corresondng to the aths (, ( m and ( n m ( ( (,, b v ( r n are used aong the ath ( n = ( (. Then ( 0 can be rewrtten as foows: J n m vaues of the decson varabes n m n m ( ( ( ( J + J m + J n m = J ( ( and resectvey when otma In [7] t was roved that f J ( s the ured tter vaue for a GPS doman then the otma buffer vaues ( (, ( ( ( ( b v v for =,,..., and the otma obectve functon vaue Q are gven as foows: ( J r n b (, = mn B (,,, {,,..., } ( ( ( J r ( ( n mn mn B,, Q = where ( 4 ( 5

9 For a PGPS doman t was shown n [7] that the vaues for as foows: ( (, b v v, {,,..., m} + + and ( ( 8 Q m are gven ( ( ( J m r ( ( n b, = mn B,, ( + ( L m + ( m + L ( ( (, B J m r ( ( n Q m = mn mn, m + + ( L m + ( m + L Smary, for SCFQ doman t was shown n [7] that for m+, m+,..., n: ( ( ( J n m r ( ( n b, = mn B,, + ( m L n m+ ( n m + L ( ( (, B J n m r ( ( n Q n m = mn mn, m n + + ( m L n m+ ( n m + L ( 6 ( 7 for {,,..., n} reduces to fndng J ( (, J ( ( m. Usng Theorem III., the vaues of J ( (, J ( ( m and J ( ( n m Hence fndng the otma souton ( b ( v, J ( ( n m sovng the foowng two equatons: and can be found by ( ( ( ( J + J m + J n m = J ( 8 where 9 s obtaned by substtutng ( b ( v, smutaneousy we roceed as foows: ( ( ( ( J r n mn mn B (,, = ( B, J m r ( ( n = mn mn, m + + ( L m + ( m + L ( B, J n m r ( ( n = mn mn, m n + + ( m L n m+ ( n m + L from ( 5, ( 6 and ( 7 nto ( 9. In order to sove ( 9 8 and ( 9

10 9 Assume frst that, mn + m mn m+ n { B ( } mn, (, ( ( ( J r n > and ( ( ( B v J m r n > + ( ( + L m ( m + L (, and ( ( ( B v J n m r n > ( m L n m+ + ( n m + L ( 0 Then ( 9 reduces to the foowng: ( ( ( ( ( ( ( ( ( J r n J m r n J n m r n = = m + n m+ ( m + L ( n m + L ( By sovng ( 8 and J ( ( n m as foows: J ( ( smutaneousy we can obtan the otma vaues of ( J = m m + n m n m L+ + L m m + J L + J ( ( m = m m + n m n m L+ + L nm n m+ J L + J ( ( n m = m m + n m n m L+ + L J (, J ( ( m and Now ut the vaues of J ( (, J ( ( n m and J ( ( m nto the nequates ( 0. If the nequates ( 0 hod then J ( (, J ( ( m and J ( ( n m are the otma tter vaues for subaths (, ( m and ( n m resectvey. In ths case the otma souton ( (, b and the otma obectve functon vaue Q ( ( n can be obtaned by usng the otma soutons n ( 5, ( 6 and ( 7. Now we consder the case when J ( (, J ( ( n m and J ( ( m, as gven by (, do not satsfy ( 0. Wthout oss of generaty assume that the frst nequaty n ( 0 s not satsfed and the rest of the two nequates are satsfed. Ths mes that there exsts,, ( ( ( ( such that ( J r n B ( v v. In ths case the maxmum reabty vaue can not be ncreased beyond Q, < ( (, B v v ( ( n =. Ths s because ncreasng the vaues of ( b ( v, ( for w not change the vaue of

11 maxmum reabty Q ( ( n snce the buffer vaue ( b ( v, v ths case agan the otma souton ( b ( v, equatons ( 5, ( 6 and 7. s hyscay bounded from above by and the otma obectve functon vaue ( ( ( B v v, 0. In Q n can be obtaned form B. Interdoman QoS Routng wth Resource Aocaton Agorthm In ths sub-secton, we resent a dynamc rogrammng agorthm to sove the nterdoman QoS routng wth resource aocaton robem. Ths agorthm uses the souton to nterdoman ath resource aocaton robem resented n sub-secton A. The agorthm s secfed n Agorthm III.. The subroutne Pre-Process deetes edges n the grah G whch have nsuffcent resources. Agorthm III. rees on the sub-routne Int-Path-Ot whch soves the otmzaton robem (. The souton to the otmzaton robem (, whch s a seca case of (, can be used wth mnor modfcatons to sove the Int-Path-Ot n the genera case. At the termnaton of V agorthm III., Q ( s, x = Q ( s, x for every node x V such that there exsts a ath from the source node s to the V destnaton node v and Q ( s, x = f no ath exsts from s to x. The otma ath can be easy constructed usng the arent nodes π V x. The comutatona comexty of the agorthm III. s gven by comutatona stes to sove Int-Path-Ot. O K V E where t taes K Agorthm III.: Interdoman QoS Routng ( G ( V, E, s,. Pre-Process ( G. For each x ad ( s J, Q Q ( s, x : = Int Path Ot ( G, x, s. For each x ad ( s π ( x : = x 4. For each x ad ( s Q ( s, x : = 5. for : = V 6. for each node x V { s } + 7. Q ( s, x : = Q ( s, x + 8. π ( x : = π ( x 9. for each node ω ad ( x 0. π π + od : = ( x +. π ( x : = ω. Tem : = Int Path Ot ( G, x, s +. f Tem > Q ( s, x + 4. Q ( s, x : = Tem 5. ese + π 6. ( x : = π od, R IV. CONCLUSION In ths aer we resented a new formuaton and souton of the Interdoman QoS routng robem. Our robem formuaton taes nto consderaton the heterogenety of the queung servce dscnes emoyed n varous autonomous systems comrsng the Internet. Our routng rotoco uses QoS metrcs whch are functons of the buffers n the routers and the n bandwdths. We rovde souton to a seca case of nterdoman QoS routng robem n a networ consstng of three autonomous systems emoyng GPS, PGPS and SCFQ queung servce dscnes. Our souton technque can be generazed to autonomous systems emoyng other queung servce dscnes n a smar manner. Our roosed formuaton can be extended by fndng a generazed nterdoman QoS routng rotoco whch can accommodate a arge number of queung servce dscnes.

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13 TABLE I DEFINITIONS OF SYMBOLS Symbo Quantty Q Mnmum reabty ured by user. Reabty s defned as the rato of the acets receved to acets transmtted aong a ath J Maxmum ured end-to-end tter deay R Mnmum ured average bandwdth Q uv Reabty of the n ( uv, J Jtter of n ( uv, uv (, b u v Buffer aocated to th data sesson n the queue resdng n node u and transmttng to node v B ( uv, Maxmum buffer sace avaabe to a data sesson transmttng from node u to v r ( u, v Bandwdth aocated to th data sesson aong the n ( uv, C( u, v Maxmum transmsson caacty of the n ( uv, C( u, v max Maxmum bandwdth avaabe to a connecton max aong n ( uv,, C( u, v C( u, v Q ( ( n Reabty of ath 0 sesson J ( ( n Jtter aong ath r ( ( n Bandwdth of ath (, n = v, v,..., vn for th data n for th data sesson n for th data sesson Aout tu Arrva functon of th data sesson at the outut queue of node u AS th Autonomous System Bucet sze for th data sesson n th Autonomous System ρ Average traffc rate for th sesson n th Autonomous System φ ( uv, Fracton of the bandwdth aong n ( uv, assgned to th data sesson L Maxmum acet sze n th Autonomous System q u, v Maxmum number of bacogged bts from th data sesson n the outut queue of node u q ( ( n Maxmum number of bacogged bts from th data sesson n the ath ( n = v0, v,..., vn K ( uv, Tota number of sessons traversng the n ( uv, Queung Servce Dscne TABLE II BANDWIDTH FUNCTIONS Bandwdth of ath ( n, r ( ( n φ v, v GPS ( ( mn n Kv (, ( φ (, = PGPS φ (, mn n Kv (, ( φ (, = SCFQ φ (, mn n Kv (, ( φ (, =

14 Queung Servce Dscne TABLE III RELIABILITY FUNCTIONS Reabty of ath ( n, Q ( ( n GPS mn { b (, } n mn, b, v mn mn, n + L b, v mn mn, n + L PGPS ( SCFQ ( Queung Servce Dscne GPS TABLE IV JITTER FUNCTIONS Jtter aong ath ( n, n mn b =,, r ( n ( J n ( ( PGPS q n r n SCFQ J q ( ( n n ( ( n = r ( ( n + ( (, C( v, v K v v L =