Centrum voor Wikunde en Informatica Performance evaluation of trategie for integration of elatic and tream traffic R. Núñez Queija, J.L. van den Berg, M.R.H. Mandje Probability, Network and lgorithm (PN) PN-R9903 February 8, 1999
Report PN-R9903 ISSN 1386-3711 CWI P.O. Box 9079 1090 GB mterdam The Netherland CWI i the National Reearch Intitute for Mathematic and Computer Science. CWI i part of the Stichting Mathematich Centrum (SMC), the Dutch foundation for promotion of mathematic and computer cience and their application. SMC i ponored by the Netherland Organization for Scientific Reearch (NWO). CWI i a member of ERCIM, the European Reearch Conortium for Informatic and Mathematic. Copyright Stichting Mathematich Centrum P.O. Box 9079, 1090 GB mterdam (NL) Kruilaan 13, 1098 SJ mterdam (NL) Telephone +31 0 59 9333 Telefax +31 0 59 199
Performance Evaluation of Strategie for Integration of Elatic and Stream Traffic Rudeindo Núñez Queija CWI P.O. Box 9079, 1090 GB mterdam, The Netherland indo@cwi.nl Han van den Berg and Michel Mandje KPN Reearch P.O. Box 1, 60 K Leidchendam, The Netherland {J.L.vandenBerg,M.R.H.Mandje}@reearch.kpn.com Current affiliation: Bell Laboratorie/Lucent Technologie, 600 Mountain ve., P.O. Box 636, Murray Hill, NJ 0797-0636, US michel@reearch.bell-lab.com BSTRCT Thi paper deal with the integration of tream traffic and elatic traffic in one ingle network, e.g. an TMbaed or an IP-baed network. Here tream traffic refer to traffic with a certain bandwidth guarantee, wherea elatic traffic flow can adapt their rate to the link bandwidth left over by the tream flow. Firt, model are developed that decribe different trategie for haring link capacitie between the tream and elatic flow. Then we give mathematical method for obtaining performance meaure, in particular call blocking probabilitie and file tranfer delay. Finally, thee method are ued for aeing and comparing the efficiency gain achieved by the integration trategie. 1991 Mathematic Subject Claification: 60K5, 68M0, 90B1, 90B. Keyword & Phrae: Integrated ervice, tream and elatic traffic, real-time and bet-effort traffic, file tranfer delay, call blocking probability, proceor haring queue. Note: The work of the firt author wa carried out in PN.1. a part of the project Quality in Future Network of the Telematic Intitute, and while he viited KPN Reearch (Leidchendam) on a part-time bai in the period March June, 1998. 1. Introduction Two major network concept have been propoed to upport large-cale multiervice network: TM (ynchronou Tranfer Mode) and IP (Internet Protocol). Originally, IP network (particularly the Internet) were built for data tranfer purpoe. Conequently, they were not appropriate for upporting real-time ervice; all traffic wa handled on a bet effort bai. For that reaon, within the Internet ociety, notably the Internet Engineering Tak Force (IETF), coniderable effort i put into concept for introducing Quality of Service (QoS) guarantee for prioritied tream, ee for intance Van der Wal et al. [] and White and Crowcroft [3]. Several propoal have been made, the merit of which are currently invetigated, particularly within the IETF working group interv and differv. In the telecommunication world, however, there i a trong impetu toward a multiervice network baed on TM, a tandardied by ITU and TM Forum. TM network have been deigned from the point of view that application require a trict QoS level. For that reaon, TM i particularly uited for upporting real-time ervice (having tringent delay requirement). In TM original form,
1. Introduction there wa no pecific facility for handling traffic with relatively low QoS requirement (for intance data tranfer), leading to an inefficient ue of network reource. In order to cope with thi problem, the development of tranfer capabilitie uch a BR (vailable Bit Rate) and UBR (Unpecified Bit Rate) tarted. the bandwidth allocated to BR and UBR trongly depend on the network congetion, there i a trong imilarity with IP bet effort cla. From the above decription, we ee that both network concept aim at integrating traffic with a certain bandwidth guarantee (or tream traffic, cf. Robert [18, 19]), and elatic traffic, that can cope with a non guaranteed, variable, bandwidth. Stream traffic mut maintain a o-called time integrity; it i generated by interactive application, like telephony and interactive video. In TM, thi tream mode i upported by the tranfer capabilitie DBR (Determinitic Bit Rate) or SBR (Statitical Bit Rate), in IP by the guaranteed QoS cla, poibly in conjunction with RSVP (ReSerVation Protocol), e.g. White and Crowcroft [3]. Elatic traffic doe not exit on it own, in that the rate at which the ource are allowed to end traffic into the network are determined by the level of network congetion. For thee elatic flow particularly emantic integrity hould be preerved. In BR thi integrity i achieved by a feedback loop (reporting the ource on the level of congetion in the network) in conjunction with a large buffer; in IP by the TCP feedback loop together with retranmiion. Thi paper aim at hedding light on the merit of the integration of tream traffic and elatic traffic. From the point of view of operational complexity, it i probably preferable to have two (or even more) dedicated network; but regarding efficient ue of reource, integration may be beneficial. It i thi efficiency gain (in term of bandwidth) achieved by integration that we ae. To get inight into the above iue, network performance analyi i required. Performance tudie on network with elatic traffic can be roughly categoried into two group: (1) Detailed tudie, mainly at cell/packet level, of the performance of BR and TCP/IP feedback mechanim. See e.g Bonomi et al. [5], Ritter [17], and Blondia and Caal [], who tudy the performance of variou BR feedback policie. The performance of everal variant of TCP/IP i tudied in e.g. Lakhman and Madhow [11]. Studie in thi category typically invetigate buffer requirement (in a bottleneck node), throughput and the impact of round trip delay on thee performance meaure. nalytical reult are motly only available for the cae of a ingle elatic traffic ource feeding into the network. Mot paper do not conider the integration with tream traffic. () Performance tudie at call level, in order to tudy the impact of the interaction between elatic traffic flow and tream traffic flow on throughput, tranfer delay and blocking probabilitie. In thee call level model, the feedback mechanim i aumed to be ideal (i.e. intantaneou feedback). With thi aumption, a network link carrying only elatic traffic flow can be modelled by a proceor haring queue. The application of proceor haring queue to tudy the performance of elatic traffic wa identified by e.g. Robert [18, 19] and Núñez Queija and Boxma [15]. The performance of proceor haring queue ha been extenively tudied and many reult (particularly on the queue length and tranfer delay ditribution) are available, ee e.g. Coffman et al. [6], Ott [16], Schaberger [0], and Yahkov [], and the urvey paper of Yahkov [5, 6]. However, we are particularly intereted in the behaviour of integrated tream and elatic traffic. In the integrated cae, the claical proceor haring queue ha to be extended in order to model the impact of the preence of a varying number of tream traffic call. Firt rough etimate for the performance of integrated tream and elatic traffic were provided by Lindberger [10]. more advanced modelling i propoed in Núñez Queija and Boxma [15], Blaabjerg et al. [3], and ltman et al. [1]. Thee tudie underly the methodology applied in the preent paper. The contribution of thi paper i twofold. In the firt place, we preent a mathematical modelling and performance analyi of the integration policie. Secondly, uing thi method, we preent an
. Problem decription 3 extenive numerical tudy in order to get inight into the amount of network reource that can be aved by different integration policie. We organied thi paper a follow. Section further pecifie the cope of the paper. In Section 3, the model i decribed and preliminary reult on the relevant performance meaure are provided. In Section we analye the elatic traffic tranfer delay in greater detail. Section 5 provide numerical reult. We draw concluion in Section 6.. Problem decription We conider a ingle network link with a certain capacity (bandwidth), that carrie both tream traffic and elatic traffic. Stream traffic conit of call requiring a given bandwidth, to be guaranteed by the network (in fact, in cae of a variable bit rate tream traffic call, thi bandwidth i the effective bandwidth). Thee call arrive according to ome tochatic proce, and are cleared after ome random time. We aume that an elatic traffic call i a file to be tranferred; the file (having a random ize) arrive according to a tochatic proce. The elatic traffic call hare the link bandwidth that i not ued by the tream traffic call. The (call-level) performance of the tream call i determined by the fraction of call being blocked. For elatic traffic, there are two relevant performance meaure: (1) The time it take to tranfer a file; we particularly concentrate on the mean tranfer time of a file of given ize. () The call blocking probability, in cae the elatic call are guaranteed a minimum bandwidth. For example, in the TM context the BR ervice category provide a Minimum Cell Rate (MCR); in IP network, minimum throughput for elatic traffic may be realied by the introduction of packet cheduling mechanim like weighted fair queueing (WFQ), in conjunction with certain flow admiion control cheme, ee e.g. Robert [18]. In thi paper we analye and compare the performance of three different policie/cenario for handling tream and elatic traffic call. Segregated cenario. In the firt place, we conider the cenario where tream and elatic traffic are handled by eparated reource. One part of the link rate i excluively dedicated to tream traffic, the other part i excluively dedicated to elatic traffic (i.e., virtually, two dedicated link are ued). Integrated cenario. In the oppoite cenario both traffic type completely hare the network reource. The rationale for thi cenario i the poibility of achieving a high utiliation. The elatic flow allowed on the link can fully exploit the bandwidth that i not ued by the tream flow. Mixed cenario. In thi cenario the link bandwidth i plit up into two part. One part can only be ued by elatic traffic flow. The other part of the link bandwidth i to be occupied by the bandwidth requirement of the tream traffic flow. n elatic (repectively tream) flow i blocked when the um of the guaranteed bandwidth become larger than the part of the link bandwidth aigned to elatic (repectively tream) traffic. Note, that in thi mixed cenario the bandwidth of the tream traffic part of the link that i not actually ued by tream traffic flow can be exploited a exce bandwidth by the elatic flow. The rationale for thi cenario i to have the benefit of efficiency gain (a in the integration cenario), but at the ame time offering call of both type a certain protection (at call level) againt each other, when call of one type generate (temporarily) a relatively large load. 3. Model decription and preliminary analyi In thi ection we preent the model that we have developed to decribe the three cenario of Section. Firt, in Section 3.1 we dicu the aumption that we make in our model. Then, in Section 3.,
3. Model decription and preliminary analyi 3.3 and 3., we eparately treat the three different policie, mentioned in Section. 3.1 Modelling aumption Throughout thi paper, we aume that requet for elatic traffic connection and tream traffic connection occur according to two mutually independent Poion procee, with intenitie λ e and λ call per econd, repectively. call of elatic traffic conit of a ingle file to be tranmitted. The mean file length i denoted by f e. Except in the egregated model of Section 3., we aume that the length of thee file are exponentially ditributed (in Section 6 we come back to thi aumption). Each file tranfer require a minimum guaranteed tranfer rate re 0, during the complete tranfer time. lo, the actual tranfer rate of an individual file can never exceed the maximum attainable tranfer rate r e +. Obviouly, r e + r e. For example, in the context of the BR ervice in TM network, r+ e i called the Peak Cell Rate (PCR) and re i the Minimum Cell Rate (MCR). Call of tream traffic require a fixed tranfer rate r > 0 over the complete duration of their holding time. gain with the exception of Section 3., we aume that thee holding time are exponentially ditributed. We denote the mean holding time by h. The fraction of blocked call of elatic and tream traffic are denoted by p e and p, repectively. In addition, for elatic traffic, we conider E[T e ], the mean file tranfer time, and E[T e (x)], the mean file tranfer time of a file of length x. Obviouly, for exponentially ditributed file length, E[T e ] = x=0 E[T e (x)] 1 f e e x/fe dx. Before proceeding, we firt introduce ome further notation. We ue the random variable X e (t), rep. X (t), to denote the number of elatic, rep. tream, traffic connection at time t 0. In teady tate we imply ue X e and X. The tate pace S of the proce (X e (t),x (t)) depend on the model conidered. The call admiion policy can be formulated a follow: Suppoe (X e (t),x (t)) = (n e,n ) S. Then, if a new elatic traffic call arrive at time t, it i accepted if (n e + 1,n ) S, and rejected otherwie. Similarly, a new tream traffic call i accepted iff (n e,n + 1) S. For notational convenience, we introduce the blocking region B e := {(n e,n ) S : (n e + 1,n ) / S} and B := {(n e,n ) S : (n e,n + 1) / S}. We define the teady-tate probabilitie, for all poible tate (n e,n ) S, π ne,n := P {X e = n e,x = n } = lim t P {X e (t) = n e,x (t) = n }. (3.1) It will be convenient to order the tate (n e,n ) lexicographically, i.e. (n e,n ) i preceded by all tate in the et {(n e,n ) S : n e < n e } {(n e,n ) S : n < n }. Throughout thi paper we ue thi ordering for the element of vector defined on the tate pace. For example, uing thi ordering on the correponding teady-tate probabilitie π ne,n, we define the teady-tate probability vector π := (π ne,n ) (ne,n ) S. In Section 3., 3.3 and 3., the difference between the three propoed policie are dicued in detail. We do not go into the iue of how to compute the teady-tate probabilitie efficiently for the integrated and mixed cenario. We only remark that, in both cae, the block tri-diagonal tructure of the generator allow for an efficient olution. The teady-tate probabilitie can for intance be computed uing the method of De Nitto Peronè and Grai [8] for generalied Quai Birth-Death procee (with ome minor modification). Performance meaure Once the teady-tate probabilitie π ne,n have been determined, we can compute the blocking prob-
3. Model decription and preliminary analyi 5 abilitie p and p e, and the mean number of elatic traffic connection E[X e ]: p = π ne,n, (n e,n ) B p e = π ne,n, (3.) (n e,n ) B e E[X e ] = n e π ne,n. (n e,n ) S By Little formula we alo have E[T e ] = E[X e ]/ (λ e (1 p e )). The lat performance meaure we conider, i E[T e (x)]. In the egregated model, E[T e (x)] i proportional to x, ee Section 3.. We preent the analyi of E[T e (x)] for both the completely integrated model and the mixed model in Section. 3. Segregated cenario In thi ection, we conider the pecial cae with no interaction between tream traffic and elatic traffic. For thi cae, the only aumption we make on the ditribution of the holding time of tream traffic call and the length of elatic traffic file, i that their firt moment exit. The link capacity C i plit into two part: C = C e + C. The capacity C e i permanently aigned to elatic traffic, and C to tream traffic. The tate pace i therefore given by S (eg) := { (n e,n ) IN 0 IN 0 : n e r e C e,n r C }. (3.3) For tream traffic thi reult in the Erlang lo model. In particular, p = (λ h ) K /K! K k=0 (λ h ) k /k!. (3.) Here, K = C /r i the maximum number of tream traffic connection. For elatic traffic, the reulting model i an M/G/1/K queue with o called generalied proceor haring (GPS) ervice dicipline. The elatic traffic connection are erved imultaneouly, each with peed r ne, where r ne depend on the total number of elatic traffic connection n e. In our cae we have for 0 n e K e, ( r ne = min r e +, C ) e, n e with K e := C e /r e the maximum number of elatic traffic connection. For thi queueing model, explicit performance reult are available in Cohen [7]. In particular, we obtain the mean file tranfer time E[T e (x)] for a file of given ize x, and the probability p e that a newly arriving file (elatic traffic flow) i blocked. Let, and φ 0 := 1. Then φ n := n j=1 1 r j, n = 1,,...,K e, p e = (λ ef e) Ke K e! φ Ke Ke, E[T (λ ef e) j e (x)] = (x/c e ) j=0 j! φ j Ke 1 (λ ef e) n n=0 n! φ n+1 Ke j=0 (λ ef e) j j! φ j. (3.5)
3. Model decription and preliminary analyi 6 Formula (3.5) how that, in the preent cae without interfering tream traffic, the mean file tranfer delay E[T e (x)] i proportional to the file ize x. Furthermore, the above reult for the mean file tranfer delay E[T e (x)], and the blocking probability p e depend on the file ize ditribution only through it mean value: The reult are inenitive to higher moment of the ditribution. When we take r e = 0 (i.e. really bet effort traffic) and r + e C e, our model for elatic traffic become the tandard M/G/1 proceor haring queue. In that cae, the above formula for the mean file tranfer delay reduce to the well known M/G/1 proceor haring reult (ee for intance Kleinrock [9, Formula.17]): λ e f e E[T e (x)] = (x/c e ). 1 λ e f e For the M/G/1 proceor haring queue, expreion have been found for the Laplace Stieltje Tranform of the ditribution of the conditional tranfer time T e (x), ee for intance Yahkov [], Ott [16], Schaberger [0], and Van den Berg and Boxma []. 3.3 Integrated cenario In the model with complete integration of the two traffic type, a new call (of any type) i accepted if the guaranteed performance i not violated for any connection. Thu, the tate pace i given by S = S (int) := { (n e,n ) IN 0 IN 0 : n e r e + n r C }. When n e > 0, the tranfer rate of each elatic traffic connection i ( r ne,n := min r e +, C n ) r. (3.6) n e i.e. the capacity available to elatic traffic i divided equally among all elatic traffic connection, but never exceeding the maximum rate r e + per connection. Our aumption on Poion arrival and exponentially ditributed file length (for elatic traffic) and holding time (for tream traffic), enure that the pair (X e (t),x (t)) i a Markov proce. Denote the maximum number of elatic traffic connection and tream traffic connection, by K e and K, repectively. Furthermore, define the maximum number of tream traffic connection when there are n e elatic traffic connection, by K (ne) := C ne r e r, n e = 0,1,...,K e. Obviouly, K = K (0). With the pair (n e,n ) ordered lexicographically, the generator of the proce (X e (t),x (t)) i given by Q (int) := Q (0) d λ e I (0) 0...... 0 M (1) Q (1) d λ e I (1) 0.... 0................ M (K e 1) Q (Ke 1) d. λ e I (Ke 1) 0... 0 M (Ke) Q (Ke) d. (3.7) Here, Q (int) conit of K e +1 block row and block column. The ize of the block are not fixed. The matrice I (ne), n e = 0,1,...,K e 1, are of dimenion (K (ne) +1) (K (ne+1) +1). It entrie are given by [I (ne) ] n,n = 1, n = 0,1,...,K (ne+1), and zero in all other poition. The dimenion of M (ne),
. nalyi of the conditional mean tranfer time 7 n e = 1,,...,K e, i (K (ne) equal to zero. Finally, the matrice Q (ne) d Except for the diagonal element, Q (ne) d M/M/K (ne) /K (ne) +1) (K (ne 1) +1), with [M (ne) ] n,n = n e r ne,n /f e, and all other entrie, n e = 0,1,...,K e, are of dimenion (K (ne) +1) (K (ne) +1). i equal to the generator of the queue length proce of the model. The diagonal element are uch that each row of Q (int) um up to 0. 3. Mixed cenario in the model with complete egregation, a fixed capacity C e > 0 i excluively reerved for elatic traffic. The remaining capacity C > 0 i primarily dedicated to tream traffic, but whenever tream traffic connection do not fill the capacity C, elatic traffic may ue the pare capacity. However, thi capacity i immediately allocated to tream traffic, a oon a a new tream traffic connection i requeted. Therefore the capacity C e hould alway be ufficient to guarantee the minimum tranfer rate re to each proceeding elatic traffic call. Hence, the tate pace of the proce (X e(t),x (t)) i the ame a for the egregated model: S (mix) = S (eg), ee (3.3). The tranfer rate r ne,n of an elatic traffic connection i, a in the integrated model, given by (3.6), with C = C e + C. Of coure, the proce (X e (t),x (t)) i again a Markov proce. in Section 3.3, we denote the maximum number of elatic traffic connection and tream traffic connection by K e and K, repectively. The number of tate in S (mix) i (K e + 1) (K + 1). Since the elatic traffic doe not affect the tream traffic, X (t) evolve a the queue length proce of the tandard Erlang lo model, jut a in the egregated model. The proce (X e (t),x (t)) i a finite inhomogeneou Quai Birth Death (QBD) proce. It generator Q (mix) ha the ame tructure a Q (int) in (3.7). However, thi time the ize of the block are all equal: The matrice I (ne), M (ne), and Q (ne) d, are of dimenion (K + 1) (K + 1). The matrice I (ne), n e = 0,1,...,K e 1 do not depend on n e, and are equal to the identity matrix. M (ne), n e = 1,,...,K e i the diagonal matrix ne f e diag[r ne,0,r ne,1,...,r ne,k ]. For convenience of notation, we et M (0) equal to the null matrix. Then, for n e = 0,1,...,K e 1, Q (ne) d = Q λ e I M (ne), and Q (Ke) d = Q M (Ke), where Q i the (tri-diagonal) infiniteimal generator of the queue length proce of the tandard Erlang lo model.. nalyi of the conditional mean tranfer time Once the teady-tate probabilitie have been determined (e.g. uing the method in De Nitto Peronè and Grai [8]), the mean ojourn time E[T e ] i eaily computed, ee the remark following (3.). However, for elatic traffic we are alo intereted in E[T e (x)], the mean tranfer time of an accepted file with given length x. Recall that, in the egregated model, E[T e (x)] i proportional to x, ee (3.5). In thi ection we analye E[T e (x)] in both the integrated and the mixed model. For detail on the analyi, for interpretation of variou entitie, and for full proof in thi ection, we refer to Núñez Queija [1]. in Section 3.1, we denote the tate pace generically by S. Thu, either S = S (int) or S = S (mix). Let S := {(n e,n ) S : n e > 0}. We retrict ourelve to the cae where r ne,n > 0 for all (n e,n ) S. Note that thi condition i automatically atified when re > 0. For the mixed trategy, the condition i alo atified when C e > 0. The cae with r ne,n = 0 for ome (n e,n ) S, can be treated in a imilar way, ee Núñez Queija [1]. For (n e,n ) S, and x 0, we introduce the following conditional expectation: β ne,n (x) = the expected tranfer time of a (non-blocked) file of length x, tarting with n e 1 other proceeding elatic traffic connection and n tream traffic connection. Let β(x) be the vector (β ne,n (x)) (ne,n ) S, where the β ne,n (x) are ordered lexicographically. Note
. nalyi of the conditional mean tranfer time 8 that we exclude blocked elatic traffic call. We may now write E[T e (x)] = 1 π ne 1,n 1 p β ne,n (x). (.1) e (n e,n ) S We now tudy the function β ne,n (x). Firt we formulate a ytem of differential equation and initial condition, from which we find the β ne,n (x). Then we how that thee function converge to a linear function, a x. Finally, we indicate how the β ne,n (x) can be evaluated numerically. Lemma.1 The function β ne,n (x), (n e,n ) S, atify the following ytem of differential equation and initial condition: r ne,n x β n e,n (x) = 1 + 1 ne,n +1λ β ne,n +1(x) + n β ne,n h 1(x) +1 ne+1,n λ e β ne+1,n (x) + n e 1 r ne,n f β ne 1,n (x) ( e 1 ne+1,n λ e + n e 1 r ne,n f + 1 ne,n +1λ + n ) β ne,n e h (x). (.) β ne,n (0) := limβ ne,n (x) = 0. (.3) x 0 Here, the indicator function 1 ne,n i 1 if (n e,n ) S, and 0 otherwie. Equivalently, we may write in matrix notation: R x β(x) = e + Q β(x), β(0) = 0. (.) In Lemma.1, e i a vector with all element equal to 1. R i the diagonal matrix, with the diagonal entrie being the lexicographically ordered r ne,n. Q i the generator of a Markov proce with a imilar tructure a Q (int) (or Q (mix) ) in (3.7). Proof of Lemma.1 We how the validity of (.) for (n e,n ) S uch that (n e + 1,n ) S and (n e,n + 1) S. In all other cae, imilar argument can be ued. By conditioning on the event that occur in a mall time interval of length, we may write, for 0: β ne,n (x) = + λ e β ne+1,n (x O( )) + n e 1 r ne,n f β ne 1,n (x O( )) e +λ β ne,n +1(x O( )) + n β ne,n h 1(x O( )) ( + 1 λ e n e 1 r ne,n f λ n ) β ne,n e h (x r ne,n ) + o( ). Rearranging term, and letting 0, we have the deired differential equation. The initial condition follow from the fact that we aumed that r ne,n > 0, for all (n e,n ) S. Therefore, once an elatic traffic call i accepted, it tranfer can tart immediately. The ytem of differential equation and initial condition in Lemma.1, uniquely determine the function β ne,n (x), x 0. The olution i given in the next theorem, ee alo Núñez Queija [1]. Theorem.1 Let π = (πn e,n ) (ne,n ) S generator Q : I.e., π Q = 0. Define, c := n e r ne,n πn e,n, (n e,n ) S p e := (n e,n ) S :(n e+1,n )/ S π n e,n. be the teady-tate ditribution vector correponding to the
5. Numerical reult 9 Let γ = (γ ne,n ) (ne,n ) S be the unique olution to, R 1 Q γ π Rγ = 0. = R 1 e 1 c λ e f e (1 p e )e, Then the unique olution to (.) i given by: β(x) = x c λ e f e (1 p e )e + [ I exp { xr 1 Q }] γ. (.5) The entitie c and p e have the following intuitive meaning: In a ytem with one permanent elatic traffic connection, c i the average capacity aigned to elatic traffic per time unit; and p e i the blocking probability of new elatic traffic call. The exitence and uniquene of γ i a well known reult from Markov deciion theory: The number γ ne,n can be interpreted a relative cot in a Markov proce with generator R 1 Q, ee for intance Tijm [1, Theorem 3.1.1 and p. 0]. Solution (.5) can be checked by ubtitution in (.). Note that, from Theorem.1 and Expreion (.1), we have an explicit expreion for E[T e (x)] in term of x. t the end of thi ection we indicate how thi expreion can be ued for computation of E[T e (x)]. Firt, however, we etablih a relevant limiting reult for β(x) and E[T e (x)] a x. Corollary. For all (n e,n ) S, lim β x n x e,n (x) c λ e f e (1 p e) = γ n e,n, and hence lim E[T x e(x)] x c λ e f e (1 p e ) = 1 π ne 1,n 1 p γ ne,n. e (n e,n ) S Corollary. follow from the fact that R 1 Q i the generator of a finite, irreducible Markov proce: It larget eigenvalue i 0, ha multiplicity 1 and correponding left null vector π and right null vector e. Numerical evaluation of the conditional mean tranfer time To compute E[T e (x)], one may ue Expreion (.5) and (.1). The term exp { xr 1 Q } γ can be evaluated in a numerically table way, by uing uniformiation: Let η > 0 be uch that P := I + 1 η R 1 Q i a non-negative matrix. Then, P i a tochatic matrix that can be aociated with the uniformied jump proce of the Markov proce governed by R 1 Q. Now, exp { xr 1 Q } = e ηx exp {ηxp} = e ηx (ηx) k P k, k! and the term in thi expreion only involve non-negative number. an alternative, we may ue (.) directly to compute recurively the coefficient of the Taylor erie of β(x) around 0. gain, thi hould be done uing P intead of Q. The advantage of thi alternative i that γ need not be computed. In Experiment 3 of Section 5, we ued both method when computing E[T e (x)]. In all cae the relative difference between the outcome of both method wa of the order of 10 8, or maller. 5. Numerical reult Uing the analyi preented in Section 3 and, we performed an extenive numerical tudy on the integration policie defined in Section. In thi ection, we preent ome of our reult. k=0
5. Numerical reult 10 Link C (all model) 155 Mbit/ C e (not for integr.) 105 80 55 30 5 Mbit/ Elatic traffic f e 50 Mbit re 0 Mbit/ r e + 10 50 155 Mbit/ Stream traffic h 10 ec. r 5 Mbit/ Table 1: Parameter in Experiment 1 C 50 75 100 15 150 λ 0.6118 0.81080 1.0306 1.6156.03378 Table : Load of tream traffic (in term of λ ) for the mixed and egregated trategie; p = 0.01 It hould be emphaied that quite a number of parameter play a role in our model. Thi, of coure, make it impoible to draw general concluion over the entire parameter pace. In order to cope with that, we fix a number of parameter at a realitic value. The guaranteed rate re for elatic traffic i taken equal to zero, in the firt three experiment. Notice that the re = 0 aumption relate e.g. to the mot likely next Internet ituation with two traffic clae: high priority (tream) traffic and low priority bet effort (elatic) traffic without any bandwidth guarantee. In the fourth experiment re > 0, which relate for example to the ituation of an TM network with BR connection having an MCR (Minimum Cell Rate) larger than zero, or to a future IP network with appropriate packet cheduling and flow admiion control mechanim in the router (ee e.g. Robert [18]). Experiment 1 In our firt experiment, we compare the efficiency of the three cenario (egregated, integrated and mixed). More preciely tated, given certain performance requirement of the two traffic type (mean file tranfer time E[T e ] for elatic traffic and blocking probability p for tream traffic call), we determine the maximum traffic load that can be handled under the three different trategie. Table 1 how the model parameter. We have choen the traffic parameter uch that the call of the two traffic type have the ame mean ize (i.e. have the ame mean number of bit to be tranferred): f e = h r. For variou value of the parameter C = C C e and r + e, we evaluated the efficiency of each of the three trategie in the following way: We have choen λ uch that the blocking probability p of the tream traffic call for the mixed and egregated trategy equal 0.01 (note that λ can be eaily computed from the Erlang lo formula), ee Table. In order to make a fair comparion, in the integrated cenario we have reduced the value of λ, uch that the amount of accepted tream traffic i equal for all three trategie. Then, given a certain load of tream traffic (in term of λ ), we determined for each of the three trategie the maximum poible load of elatic traffic (in term of λ e ), uch that E[T e ] = h = 10. The reult of thi firt experiment are hown in Figure 1. For C e = 5 the allowed λ e i maller than 10 5. expected, the mixed and integrated trategie are coniderably more efficient than the egregated trategy: apparently, the elatic traffic benefit highly from the fluctuating amount of bandwidth that i left over by the tream traffic. The difference between the mixed trategy and the integrated trategy are very mall. In all cae, the mixed trategy i at leat a efficient a the integrated trategy. Finally it i noted that the impact of r + e on the efficiency of the trategie i very mall. Thi i due
5. Numerical reult 11 3 e.5 1.5 1 0.5 0 C e = 105 egregated mixed integrated C e = 80 C e = 55 C e = 30 C e = 5 155 50 10 155 50 10 155 50 10 155 50 10 155 50 10 r + e Figure 1: Efficiency of the three trategie (in term of λ e ) to the fact that the ytem i highly loaded: the number of elatic traffic call imultaneouly preent in the ytem i mot of the time that large, that each of them receive le than 10 Mbit/ of the total available capacity (hence, it make no difference whether r e + = 10,50 or 155 Mbit/). Experiment In the previou experiment, tream traffic call and elatic traffic call arrive/depart at more or le the ame time cale. What if thi i not the cae, i.e. what if the tream traffic fluctuate much fater or much lower than the elatic traffic? To invetigate thi, we repeated Experiment 1 for the cae h = 1 (rapidly fluctuating tream traffic) and h = 100 (lowly fluctuating tream traffic). ll other parameter in Table 1 remain unchanged. Note that the value of λ in Table are multiplied by a factor 10 (in cae h = 1), and by a factor 0.1 (in cae h = 100), uch that p remain equal to 0.01 in the mixed and egregated trategie. We oberved that in all cae the egregated trategy i the leat efficient, and that the mixed trategy outperform the integrated trategy (particularly when h = 100). For the mixed trategy, being the mot efficient in all cae, the impact of the time cale difference i reported in Figure. Intuitively, one expect that when the tream traffic fluctuate very fat, the performance of elatic traffic i the ame a for the egregated cenario with C e equal to the mean available capacity C λ (1 p )h r. In Figure, alo the value of λ e are given for that cae. Thi phenomenon wa already noted in Núñez Queija and Boxma [15] and ltman et al. [1], and formally proved in Núñez Queija [13]. The numerical reult how that λ e increae when the tream traffic fluctuate fater (i.e. h become maller). Note that, a expected, the difference between the mixed cenario with h = 1 (i.e. tream traffic fluctuate relatively fat) and the egregated cenario with C e = C λ (1 p )h r i negligible. in the previou experiment, it i een that the impact of r + e on the efficiency i very mall. Experiment 3 For the mixed trategy, we conider the conditional mean file tranfer time E[T e (x)] a a function of the file ize x. In particular, we are intereted in how fat E[T e (x)] converge to it linear aymptote (a x ). The parameter f e, r e, h and r are fixed at their repective value given in Table 1, and C e i et equal to 80 (therefore the condition r ne,n > 0 in Section i atified). The value of
5. Numerical reult 1 e 3.5 1.5 1 0.5 0 B B C e = 105 B B C e = 80 C e = 55 155 50 10 155 50 10 155 50 10 155 50 10 155 50 10 B B C e = 30 B B h = 100 h = 10 h = 1 h # 0 B B C e = 5 r + e Figure : Efficiency of the mixed trategy (in term of λ e ) on different time cale λ (0.81) i again choen uch that p = 0.01, and λ e i fixed at.17, which i the value computed in Experiment 1 with r e + =. In Figure 3, E[T e (x)] i given for the three value of r e +. We oberve that E[T e (x)] i coniderably maller for larger value of r e +. We alo computed the aymptote of E[T e(x)]; for r e + = 10 the reult are hown in Figure. For the other two value of r+ e, we obtained imilar figure, the ditance between the actual curve and the aymptote being larger for larger r e +. Keeping λ e fixed, we repeated the above experiment for rapidly fluctuating tream traffic (h = 1) and for lowly varying tream traffic (h = 100). in Experiment, the value of λ when h = 1 (rep. h = 100) i found by multiplication by a factor 10 (rep. 0.1), uch that the traffic load of tream traffic (in term of λ h ) i the ame in all cae. In both cae, the reult yield graph (not hown in thi paper) imilar to the one in Figure 3 and. However, for fat tream traffic we oberved that the ditance between E[T e (x)] and it aymptote i coniderably maller, and that for low tream traffic thi ditance i very large. The reult how that in general the aymptote doe not give a ueful approximation for E[T e (x)]. n additional numerical tudy indicate that a good approximation i provided by the tangent of the curve in the origin. In Figure, for value of x maller than five time the mean file ize f e = 50 Mbit, the relative difference between E[T e (x)] and the tangent in zero i le than.5%. Note that the lope of thi tangent line can be eaily computed from the teady-tate ditribution, ee Núñez Queija [1]. Experiment In our lat experiment we conider the ituation that the elatic traffic call are guaranteed a certain minimum bandwidth re. For the mixed trategy, we tudy the impact of C on the call blocking probabilitie p e and p of the elatic traffic and the tream traffic, repectively. before, we chooe f e = 50 Mbit and r = 5 Mbit/. Furthermore, h = 10 ec., re = 5 Mbit/ (i.e. the tranfer time of a file of ize x Mbit i bounded by x/5 econd), and r e + = 155 Mbit/. We fix the call arrival intenitie at λ e = 1.90 and λ = 1.15. Thee value are choen uch that p e = p = 0.05 in the mixed cenario with C e = 75 Mbit/. The reult are hown in Figure 5. It i een that the call blocking probability for the tream traffic decreae very rapidly when C increae, while the blocking probability for the elatic traffic grow only moderately. Note that, a C increae, the amount of bandwidth (C e = C C ) reerved for elatic traffic decreae. part of thi reaigned bandwidth i
5. Numerical reult 13 30 E[T e (x)] 5 3 0 3 3 15 3 3 3 r + e = 155 3 10 r + e = 50 3 r + e = 10 3 5 3 3 0 0 10 0 30 0 50 60 70 80 90 100 110 10 Figure 3: Conditional mean delay for h = 10, and r + e = 10,50,155 x 50 E[T e (x)] 00 150 100 50 E[T e (x)] aymptote tangent in origin 0 0 100 00 300 00 500 600 700 800 900 1000 x Figure : E[T e (x)] and it aymptote for h = 10 and r + e = 10
6. Concluion and direction for future reearch 1 0. 0.18 0.16 0.1 0.1 0.1 0.08 0.06 0.0 0.0 3 3 3 3 p e (mixed) p e (egregated) p 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 10 0 30 0 50 60 70 80 90 100 110 10 130 10 150 C Figure 5: Blocking probabilitie for different choice of C however not ued by the tream traffic. Thi amount of bandwidth, C λ (1 p )h r, allocated to, but not ued by the tream traffic, i apparently very well exploited by the elatic traffic call. Thi i confirmed by the reult for the lo probability of elatic traffic call in the correponding egregated cae, which are alo hown in the figure. 6. Concluion and direction for future reearch In thi paper we tudied the integration of tream traffic and elatic traffic in one ingle network, e.g. an TM-baed or an IP-baed network. Firt, model were developed decribing different integration trategie. Then we preented analytical technique for obtaining performance meaure, in particular call blocking probabilitie and file tranfer delay. Finally, thee method were ued for aeing and comparing the efficiency gain achieved by the integration trategie. Integration of tream and elatic traffic The firt concluion i that integration of tream and elatic traffic in one ingle network i much more efficient (with repect to the ue of network reource) than having two dedicated network for the two traffic type (i.e. egregation). The o-called mixed cenario i lightly more efficient than the integrated cenario, and ha the additional advantage of offering call of both type a certain protection againt each other, when call of one type generate (temporarily) a relatively large load. For other integration cheme like trunk reervation the analyi and computation of the performance meaure can be done in a imilar way; comparion with the integration trategie conidered in thi paper would be an intereting iue for further reearch. nalytical technique We demontrated that the relevant performance meaure can be analyed and efficiently calculated in a numerically table way. In particular, we developed a technique for evaluating the mean tranfer time E[T e (x)] of an elatic file of given length x. Our numerical tudy howed that, for value of x up to four or five time the mean file ize, a good approximation of E[T e (x)] i provided by the tangent line in the origin; the lope of thi tangent line can be eaily determined from the teady-tate ditribution. poible direction for further reearch i the following. In the preent tudy file length are (motly) aumed to have an exponential ditribution. Thi aumption allowed for a detailed analyi of the impact of the interaction between both traffic type on their performance. Extenion of our analyi to phae type ditribution i poible and we expect that imilar reult hold. However,
6. Concluion and direction for future reearch 15 extenion to the cae of file ize ditribution with heavy tail, e.g., the Pareto ditribution, i not traightforward. It would be ueful to be able to compute the relevant performance meaure under thi modelling aumption, cf. Zwart and Boxma [7] for the cae that only elatic traffic call hare the link bandwidth. n intereting quetion then i whether our concluion regarding integration/egregation till hold. In particular, can E[T e (x)] (for quite large value of x) till be approximated by it tangent in the origin and doe it converge to a linear function when x grow to infinity? cknowledgement The author would like to thank S.C. Bort, O.J. Boxma, R.E. Kooij, and.p. Zwart for carefully reading previou verion, and for providing comment that led to improvement of the paper.
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