A Sub-Critical Deficit Round-Robin Scheduler

Transcription

1 A Sub-Crtcal Defct ound-obn Scheduler Anton Kos, Sašo Tomažč Unversty of Ljubljana, Faculty of Electrcal Engneerng, Ljubljana, Slovena E-mal: Abstract - A scheduler s an essental element of a network devce. Network devce s capabltes n a great deal depend on the propertes of ts scheduler and the schedulng algorthm used. The most mportant measures descrbng a schedulng algorthm are: latency, farness and complexty. Ths paper dscuses a new schedulng algorthm called Sub-Crtcal Defct ound- obn SCD), together wth ts latency, farness and complexty measures. Delay s an mportant transmsson parameter. A sgnfcant part of an end-to-end delay of a data flow s latency the delay nduced by schedulng algorthms n network devces. In ths paper we present a derved latency of the SCD scheduler. We compare the latency of the SCD wth latences of some other well-known schedulers. Farness s also a much desred property of schedulng algorthms. SC- D scheduler provdes farness for flows wth varable packet lengths, and allows bandwdth reservaton. In ths paper we present a derved absolute and relatve farness measures for the SCD scheduler. We also compare them wth farness measures of some other well-known schedulers. The complexty of a schedulng algorthm should be as low as possble. We wll show that the complexty of the SCD scheduler s constant. We present analytcal and smulaton results for the complexty of SCD. 1 INTODUCTION In packet networks wth statstcal multplexng lke nternet) overload causes congeston that s solved ether by delayng or by droppng excess packets. Many of the problems that we face n networks are related to the allocaton of a lmted amount of shared resources buffer, memory, bandwdth, etc.) to competng data flows. There are dfferent solutons that try to solve a challenge of assurng hgh resource utlzaton and hgh applcaton performance at the same tme. Essentally they can be grouped nto two categores: end-system based solutons and network based solutons. In ths paper we analyse network based solutons. Many multmeda applcatons rely on the ablty of the network to provde some sort of qualty of servce guarantees. The term Qualty of Servce QoS) can generally be defned as a set of network mechansms that satsfy the vared qualty of servce levels requred by applcatons, whle at the same tme maxmzng bandwdth utlzaton. Applcatons rely on traffc schedulng algorthms n swtches and routers to guarantee performance bounds and meet the agreed QoS. There are several measures that are to be consdered when choosng a schedulng algorthm. The most mportant are: farness, latency and complexty. A good schedulng algorthm should always provde the best possbltes for provdng the desred QoS. It has to be far, t has to have a bounded maxmum delay lmt, ts complexty should be low, t should have easy mplementaton, and hgh effcency. In ths paper we are analysng the performance of the SCD scheduler n the terms of the above gven measures. The paper s organsed as follows. In Sectons 2 to we brefly descrbe why latency, farness and complexty are so mportant propertes of each scheduler. We contnue wth a very short presentaton of some wellknown schedulers n Secton 5 and wth the explanaton of the bascs of SCD scheduler n Secton 6. In Sectons 7 to 9 we present the derved measures for the latency, farness and complexty of the SCD scheduler. 2 IMPOTANCE OF LATENCY The end-to-end delay s a very mportant QoS parameter. A number of factors contrbute to the end-to-end delay: forwardng delay, queung delay, propagaton delay and seralzaton delay. When schedulng algorthms are dscussed, t s only the queung delay that s of our nterest. It denotes the amount of tme that a packet has to wat n a queue as the system performs statstcal multplexng and whle other packets are servced before t can be transmtted on the output port. The schedulng algorthm of a scheduler should provde end-to-end delay guarantees for ndvdual flows wthout severely under-utlzng network resources. Whle queung delay can be vewed prmarly as a delay parameter of a packets), latency s a delay parameter Tradtonally, a flow s defned as a sequence of packets generated by the same source and headed toward the same destnaton va the same path n the network. It s assumed that packets belongng to dfferent flows are queued separately whle they awat transmsson. A scheduler dequeues packets from these queues and forwards them for transmsson.

2 assocated to data flows. The noton of latency, that s gong to be used here, s based on the length of tme t takes a new flow to begn recevng servce at ts reserved rate for detals see [6]). Therefore, latency drectly affects the sze of the playback buffers requred n real-tme applcatons. 3 IMPOTANCE OF FAINESS Packets belongng to dfferent flows often share lnks n ther transmsson paths. Farness s a very desrable property n the allocaton of bandwdth on such lnks. In a multuser/multapplcaton envronment the protecton guaranteed by far schedulng mproves the solaton between flows. Isolaton offers more predctable performance of the system to users applcatons. Far allocaton of bandwdth ensures that the performance of one flow s not affected when another, possbly msbehavng flow, tres to send packets faster than ts reserved rate. In addton, strateges and algorthms for far management of network traffc can serve as a crtcal component of QoS mechansms to acheve certan guaranteed servces such as delay bounds and mnmum bandwdths. 4 IMPOTANCE OF COMPLEXITY Schedulers, partcularly n hgh-speed networks, should be as smple and modest as possble. A good scheduler does not use a lot of memory resources and s easly mplementable, not only n software, but also n hardware. The most mportant property of a scheduler, n terms of complexty, s ts computatonal complexty. It s defned as a maxmal workload number of operatons) necessary to enqueue or to dequeue a packet. It s denoted by Ox), where x represents the relaton between the number of queues and the number of operatons needed to queue or dequeue a packet. For nstance, OlogN)) means that the number of operatons s logarthmcally dependent on the number of queues N. In core networks t s hghly desrable that the complexty s O1), what means that the number of operatons s constant, regardless of N. 5 SOME WELL-KNOWN SCHEDULES Schedulng algorthms can be broadly classfed nto two categores: sorted prorty schedulers and framebased schedulers. Sorted prorty schedulers mantan a global varable called the vrtual tme or system potental functon. The prorty of each packet, called the tme-stamp, s calculated based on ths varable. The packets are then scheduled n an ncreasng order of ther tme-stamps. Examples of sorted prorty schedulers are Weghted Far Queung WFQ), Self-clocked Far Queung SCFQ), Start-tme Far Queung SFQ) and Worst-case Far Queung WF2Q). Generally, they gve good farness and low latency bound but they have great computatonal complexty. In frame-based schedulers, tme s splt nto frames of fxed or varable length. eservatons are made n terms of the maxmum amount of traffc the flow s allowed to transmt durng a frame perod. The servce receved by a flow n one round-robn opportunty s proportonal to ts far share of the bandwdth. These schedulers do not have to perform sortng among packets and calculate global vrtual tme functon, so they have lower computatonal complexty than the sorted prorty schedulers do. Defct ound- obn D), Surplus Defct ound-obn, Elastc ound-obn, Nested ound-obn are some of the frame based schedulers wth complexty O1), but they have worse farness and latency propertes than the sorted prorty schedulers. In 1996, Shreedhar and Varghese [5] proposed D, one of the most popular frame-based schedulng algorthms. The man characterstc of all D-lke schedulng algorthms s ther ablty to provde guaranteed servce rates for each flow queue). D servces flows n a strct round-robn order. It has complexty O1) and t s easy to mplement. Its latency s comparable to other frame-based schedulers. 6 SCD DEFINED SCD s a frame-based schedulng algorthm. It s a dervaton of the orgnal D schedulng algorthm. SCD has several advantages over the orgnal algorthm. In contrary to D, t allows that the amount of servce each flow queue) receves n one round-robn opportunty s less than the maxmum length of the packet of each flow. The consequences of ths concesson are that SCD has lower latency bound and better farness measure than the orgnal algorthm. We show that n sectons 7 and 8. The downsde of ths change s, that the number of operatons needed to dequeue a packet ncreases. Ths means that the computatonal complexty also ncreases. In secton 9 we show, that despte the ncrease n the number of operatons, the complexty of the SCD scheduler s stll constant OC). In ths paper we present only the end results for the latency, farness and complexty of the SCD scheduler. An exhaustve analyss wth the detaled dervaton of the presented results can be found n [3], partal analyss can be found n [1], [2], and [4].

3 Below s a lst of varables used n the followng sectons: transmsson rate of an output lnk, N total number of actve flows, r reserved rate of flow, w weght assgned to flow, Q quantum assgned to flow, F frame sze, maxmum possble packet sze. Because all flows share the same output lnk, a necessary constrant s that the sum of all reserved rates must be less or equal to the transmsson rate of the output lnk: r 1) Let r mn be the smallest of r : r mn = mn r. Each flow s assgned a weght that s gven by: w = r r mn 2) Note that 1, 2,,N holds w 1. Each flow s assgned a quantum of Q bts, that s a whole postve value,.e. Q N. Ths quantum s actually the amount of servce that the flow should receve durng each round-robn servce opportunty. Let us defne wth Q mn the mnmum of all the quanta. Then the quantum for each flow s expressed as: Q = w Q mn 3) Frame sze F s the sum of all the quanta Q and s gven by the expresson: latency s based on the length of tme t takes a new flow to begn recevng servce at ts guaranteed rate. Usng the general dea of Stlads and Varma n [6] we derve the upper latency bound θ for the SCD algorthm. It s gven by the expresson: 1 θ Q 1 ) + r Q + 1) N ) 5) Fr r A more detaled analyss s gven n [3], [4], and [1]. We compare the latency of the SCD wth latences of some other well-known schedulers n Table 1. 8 FAINESS OF SCD When there s contenton for resources, t s mportant for resources to be allocated farly. Among schedulng algorthms sgnfcant dscrepances may exst n servce provded to dfferent flows over the short term. For example, two schedulng algorthms may have the same delay guarantees but can have very dfferent farness behavours. There s no commonly accepted method for estmatng the farness of a schedulng algorthm. In general, we would lke the system to always serve flows proportonal to ther reserved rate and dstrbute the unused bandwdth left behnd by dle flows proportonally among actve ones. In addton, flows should not be penalzed for excess bandwdth they receved whle other flows were dle. F = Q 4) For an arbtrary nterval n tme and n any executon of SCD servce dscplne the FM s gven by the nequalty: 7 LATENCY OF SCD Stlads and Varma n [6] defned a general class of schedulers, called Latency-ate L) servers. The behavour of a L server s determned by two parameters the latency and the allocated rate. Latency of a L server s the worst-case delay seen by the frst packet of a busy flow. That s, the packet arrvng when the flow s queue s empty. The latency of a partcular schedulng algorthm may depend on ts nternal parameters: ts transmsson rate on outgong lnk, the number of flows sharng the lnk and ther allocated rates. The authors also developed and defned the noton of latency of the schedulng algorthm and determned an upper bound on the latency for a number of schedulers that belong to a class of L servers. Ths noton of F M w + + Q mn 1 w 1 6) Our objectve s to fnd the maxmum value of the expresson on the rght hand sde of nequalty 6) and along wth that the upper bound of FM. Snce our nterest s only to fnd the local maxmum, we can proceed by analysng every varable separately. s defned by system propertes, so we can treat t as a constant. Q mn ) w So the expresson 7) has ts maxmum when both w and take ther mnmal possble value that s 1. We can conclude that for a gven Q mn, the upper bound of FM occurs when at least two flows have ther quanta equal to Q mn.

4 Scheduler Latency bound elatve farness GPS 0 0 FCFS WFQ + L { max r max,j max C j + w { C = mn N 1) w VCFQ SCFQ FBFQ + r N 1) + r + r F Q E )+ 1)N 1) 3F 2Q D SCD Q r 1 1 )+ w 1) 1r 2 + Q ) Fr N),C + + L }) w }, max L n 1 n N w n L w { 2F Q max + L w, 2F Q, w, + L } w L w + w Q mn + Q mn 1 w 1 Table 1. Comparson of latency bounds and relatve farness measures among dfferent schedulers. Denotatons used: transmsson rate of an output lnk ), total number of actve flows N), reserved rate of flow r ), weght assgned to each flow w ), quantum assgned to flow Q ), frame sze F ), maxmum packet length ). Abbrevatons used: GPS Generalsed Packet Server), FCFS Frst Come Frst Served), WFQ Weghted Far Queueng), VCFQ Vrtual Clock Far Queueng), SCFQ Self-Clocked Far Queueng), FBFQ Frame-Based Far Queueng), E Elastc ound-obn). Absolute farness measure s ntutvely closer to us because t shows the dfference between the farness of the scheduler beng analysed and the farness of the deal GPS scheduler. We use the relatonshp between the FM and AFM derved n [7]. AF M 1 Q mn F ) F M 8) AFM for an arbtrary nterval n tme and n any executon of SCD servce dscplne s therefore gven by the expresson: AF M 1 Q ) [ mn F w ] + Q mn 1 1 w + 9) When Q mn s large the absolute farness can be low even f the correspondng relatve farness measure s hgh. We have already dscussed relatve farness bound and ts worst and best case, what s left to be analysed s the factor 1 Q mn ). Its mnmum value s beng F reached when Q mn reaches ts maxmum value,.e. F when Q mn reaches ts maxmum value, snce frame sze F s a constant. Q mn takes ts maxmum value when all the quanta are equal. In that case AFM s the smallest and the furthest from FM. We compare the FM of the SCD wth FMs of some other well-known schedulers n Table 1. 9 COMPLEXITY OF SCD The analyss of the computatonal complexty of SC- D s based on the computatonal complexty as seen by the packet watng at the head of the queue. We gve the computatonal complexty of SCD as the factor of computatonal complexty of the orgnal D algorthm. The lst of varables used: gl) Gl) L l L mn M m N m O m n p m,n L m,n DC DC m Q probablty densty functon of packet lengths cumulatve probablty functon average packet length packet length mnmum packet length maxmum packet length no. of packets arrved to the actve queue seral number of the packet n actve queue maxmum no. of rounds to dequeue packet m average no. of operatons to dequeue packet m no. of rounds from the last dequeue operaton probablty that packet m s served n round n average length of packet m served n round n current value of the Defct Counter average DC value after dequeueng packet m quantum of the queue The analyss takes nto account only one flow or queue of the SCD scheduler. All varables defned above are referrng to the same queue. The ntal condtons are:

5 DC 0 = 0 the ntal value of DC s by defnton always 0. Ths s the value of DC at the tme the frst packet arrves nto the queue and the queue becomes actve. L mn Q 1 the quantum Q should have a value between L mn and. Values below L mn do not make any sense as SCD would need more than one round to dequeue even the shortest packet. Values above would mean that SCD s operatng as the orgnal D algorthm. 9.1 Analytcal results The detaled dervaton of the followng expressons s gven n [3]. Due to the lack of space, only the end results are lsted here. m = 1, 2, 3,...M n = 0, 1, 2,...N m Lmax DC m 1 N m = Q p m,n = L m,n = DC m 1+nQ DC m 1+n 1)Q gl)dl ) = GDC m 1 + nq) G DC m 1 + nq Q nq+dc m 1 n 1)Q+DC m 1 nq+dc m 1 lgl)dl gl)dl The fnal results of dervaton are O m and OM). The later represents the complexty ncrease factor between SCD and D. We gve some examples n the followng subsectons where we present the above results usng the exponental probablty dstrbuton of packet lengths. 9.2 Complexty of SCD at Exponental Dstrbuton of Packet Lengths For the exponental probablty dstrbuton we have: gl) = λe λl L mn =0 L = 1 λ Gl)=1 e λl = N m = Where 1/λ denotes the average packet length of the exponental dstrbuton. Consderng the ntal condtons, we see that we are always operatng n a subcrtcal area. The value of the quantum Q s always less than the maxmum packet length. The complete dervaton of the results s gven n [3]. Here we lst only the end results. For DC 0 =0, m =1, 2, 3,...M and n =0, 1, 2,... we have: DC m = DC m 1 1 λ + QeλQ e λdcm 1 O m = 1+ e λdcm 1 e λq 1 e λq 1 12) 13) For the calculaton of O m and Om) we have to teratvely calculate the values of DC m accordng to the expresson 12). The example for Q = 1000 s shown n Fgure 1. n 1)Q+DC m 1 dc m,n = DC m 1 + nq L m,n DC m = N m n=0 p m,n DC m 1 + nq L m,n ) From the above expressons we can calculate O m, the number of operatons needed to dequeue the m-th packet: O m = p m,0 + N m n=1 np m,n 10) Fnally we calculate OM), the average number of operatons needed to dequeue a packet from the actve queue wth M packets. OM) = 1 M M O m 11) m=1 All of the results above are general and they hold for an arbtrary probablty dstrbuton of packet lengths. Fgure 1. Comparson of O m and Om) for Q = When the queue s constantly backlogged stays actve for a long tme) we can wrte approxmate results. In ths case we can say that m and get the approxmate non-teratve result for DC m. [ DC m = 1 λ ln 1 1 e λq)] 14) λq Maxmum packet length of the exponental dstrbuton s nfnte = )

6 For small values of m we can use the value of DC 1 : QeλQ DC m = DC 1 = e λq 1 1 λ 15) The relatve dfference between the accurate teratve values and both approxmatons s gven n Fgure 2. Sm Analytcal OM) Q M =10 M = 100 M = ,060 10,093 10,053 10, ,215 2,240 2,216 2, ,369 1,385 1,370 1, ,067 1,073 1,068 1, ,005 1,005 1,005 1, Table 2. Value of OM) for SCD scheduler and exponentally dstrbuted packet lengths wth mean Concluson Fgure 2. The relatve dfference of O m between the accurate teratve values and both approxmatons at Q = From the above results t s evdent that O m and OM) reach ther maxmal values at m = 1 and for a small values of M. Small values of M are typcal for a lghtly loaded network where queues alternate between actve and passve state frequently. In each actve state there are only a small number of packets to be served. Wth ncreasng M queues begn to fll, queue actve perods become longer, O m and OM) are decreasng and approachng ther asymptotc values. Ths result s favourable for us as computatonal complexty s more mportant n a heavly loaded network. 10 Smulaton esults Let us show that analytcal and smulaton results for the SCD scheduler match. We smulate the operaton od SCD scheduler at dfferent values of Q. Packet lengths are exponentally dstrbuted wth mean 1000, quantum values are Q = 100, 500, 1000, 2000, 4000, Each smulaton comprses of one mllon packets. esults are shown n Table 2. From Table 2 we can see that OM) does not ncrease lnearly wth the decrease of quantum value. For nstance, at Q = 1000 the factor of complexty ncrease between SCD and D s only D would need Q = to be able to serve the maxmum packet length n the smulaton) and not 16. For a 16 tmes smaller Q we are penalsed only wth tmes n complexty ncrease. In ths paper we present our derved results for the latency, farness and complexty of the SCD scheduler. Snce SCD uses sub-crtcal values of Q, t can be easly shown that SCD scheduler has better latency bounds and better farness measures than the orgnal D scheduler. Ths mprovement s pad by the ncrease n computatonal complexty. Wth SCD we have the freedom to decrease the values of Q and mprove the latency of the scheduler. Wth ths we also decrease the total end-to-end packet delay and n ths way contrbute to a better operaton of network, flows, applcatons, etc. The same holds for farness. By choosng the parameters of SCD we can fne tune the scheduler to the needs of traffc on a network and n that way contrbute to the overall QoS. 12 eferences [1] Anton Kos, Sašo Tomažč: A More Precse Latency Bound of Defct ound-obn Scheduler, Elektrotehnšk vestnk, 2009, vol. 76, no. 5, p [2] Anton Kos, Sašo Tomažč: More Precse Farness Bounds of Defct ound obn Scheduler, IPSI BGD Trans. Advanced esearch, Jan. 2010, vol. 6, no. 1, p [3] Anton Kos: Zagotavljanje razlčnh stopenj kakovost stortve v omrežjh s paketnm prenosom podatkov, Doctoral Thess, Faculty of Electrcal Engneerng, Unversty of Ljubljana, Slovena, February 2006 [4] Anton Kos, Jelena Mletć: Sub Crtcal Defct ound obn, Techncal eport, Faculty of Electrcal Engeneerng, Unversty of Ljubljana, Slovena, July 2005 [5] M. Shreedhar, George Varghese: Effcent Far Queung Usng Defct ound obn, IEEE/ACM Transactons on Networkng, Volume 4, Issue 3, June 1996 [6] D. Stlads, A.Varma: Latency-ate Servers - A General Model for Analyss of Traffc Schedulng Algorthms, IEEE/ACM Transactons on Networkng, Volume 6, Issue 5, October 1998 [7] Yunka Zhou, Harsh Sethu: On the elatonshp Between Absolute and elatve Farness Bounds, IEEE Communcatons Letters, Vol. 6, No. 1, January 2002