Design and evaluation of a connection management mechanism for an ATM-based connectionless service

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1 Distributed Systems Engineering Design and evaluation of a onnetion management mehanism for an ATM-based onnetionless servie To ite this artile: Geert Heijenk and Boudewijn R Haverkort 1996 Distrib Syst Engng 3 53 View the artile online for updates and enhanements Related ontent - Quality of servie management using generi modelling and monitoring tehniques Leonard J N Franken and Boudewijn R H M Haverkort - Measures of self-bloking system with infinite spae Jyh-Bin Ke and Yu-Li Tsai - Numerial performability evaluation of a group multiast protool Luai M Malhis, William H Sanders and Rihard D Shlihting Reent itations - CSL model heking algorithms for QBDs Anne Remke et al This ontent was downloaded from IP address on 07/01/2018 at 18:21

2 Distrib Syst Engng 3 (1996) Printed in the UK Design and evaluation of a onnetion management mehanism for an ATM-based onnetionless servie Geert Heijenk and Boudewijn R Haverkort Centre for Telematis and Information Tehnology, University of Twente, PO Box 217, 7500 AE, Enshede, The Netherlands Abstrat The Asynhronous Transfer Mode (ATM) has been developed as a onnetion-oriented tehnique for the transfer of fixed-size ells over high-speed networks Many appliations, however, require a onnetionless network servie In order to provide suh a tehnique, one an built a onnetionless servie on top of the onnetion-oriented servie In doing so, the issue of onnetion management omes into play In this paper we propose a new onnetion management mehanism that provides for low bandwidth usage (as ompared to a permanent onnetion) and low delays (as ompared to a onnetion-per-paket approah) We model the new mehanism under two workload senarios: an ordinary Poisson proess and an interrupted Poisson proess We use Markovian tehniques as well as matrix-geometri methods to evaluate the new onnetion management mehanism From the evaluations it turns out that the proposed mehanism is superior to older approahes (whih an be seen as limiting ases) 1 Introdution The Asynhronous Transfer Mode (ATM) is beoming inreasingly important in teleommuniation networks ATM has been developed as a onnetion-oriented tehnique, whih implies that a onnetion should be established prior to any information transfer In order to be able to support appliations with a onnetionless nature suh as eletroni mail and information retrieval, there is a need to provide a onnetionless servie with networks based on ATM [1] Furthermore, a onnetionless servie would be very suitable for the interonnetion of the installed base of basially onnetionless Loal Area Networks (LANs) One of the key funtions in providing a onnetionless servie using ATM is onnetion management This funtion is responsible for establishing and releasing ATM onnetions in suh a way that (onnetionless) pakets an be transferred from their soure to the proper destination using these onnetions A problem appears in the fat that the onnetion management funtion has no advane knowledge about the offered traffi that has to be transferred, beause of the onnetionless nature of this traffi It should, however, make an agreement with Present address: Erisson Business Mobile Networks BV, PO Box 645, 7500 AP, Enshede, The Netherlands ( address: gehe@ensaeerissonse) Present address: RWTH-Aahen, LFG Informatik/Verteilte Systeme, D , Germany ( address: haverkort@informatikrwth-aahende) the signalling system of the ATM network about the harateristis of the reserved bandwidth for the established onnetions (eg mean and peak bandwidth) In this paper we present and evaluate the performane of a onnetion management mehanism alled On-demand Connetion with Delayed Release (OCDR) The OCDR mehanism establishes a onnetion between two nodes in an ATM network as soon as a paket needs to be transferred between these nodes After the paket has been transferred, the onnetion is not immediately released It an be used for subsequent pakets for the same destination as well Only if no paket is transferred for a ertain period of time (the holding time) is the onnetion released again We present a number of different Markov models for the OCDR mehanism The first one is a simple model, where the pakets are assumed to arrive aording to a Poisson proess Furthermore, the time needed for establishing a onnetion and the holding time of the mehanism are assumed to have an exponential distribution For this model, we present an analytial solution The seond model is a refinement of the first one, where pakets arrive aording to an interrupted Poisson proess (IPP) This model is further refined, so that the onnetion establishment time and the holding time an be modelled with an Erlang distribution The seond and third model are solved numerially using the matrix geometri solution method developed by Neuts [2], and supported by the pakage Xmgm [3] This paper is organized as follows First, in setion 2, /96/ $ The British Computer Soiety, The Institution of Eletrial Engineers and IOP Publishing Ltd 53

3 G Heijenk and B R Haverkort we define the onnetion management funtion, present alternatives to implement this funtion, and present the OCDR mehanism in more detail In setion 3, we model and analyse the performane of the OCDR mehanism under various traffi harateristis In setion 4, we evaluate the performane of the mehanism, and ompare it to the performane of some onventional strategies Finally, in setion 5, we give some onluding remarks end-system A ATM swith A CLS A ATM swith C ATM swith B CLS B end-system B 2 Connetion management In order to avoid onfusion onerning the meaning of the term onnetion management, we explain what this funtion omprises in setion 21 After that, in setion 22, we propose a andidate mehanism for onnetion management, and relate it to other, more onventional mehanisms Finally, in setion 23, a number of performane measures are identified, whih will be used to evaluate and ompare the mehanisms 21 Definition Connetion management is the funtion that interats with the signalling system of the ATM network to ensure that an ATM onnetion to the proper destination, and with suffiient bandwidth, is available for the transfer of a paket The onnetion management funtion only interats with the signalling system loally It an ask the signalling system to establish a onnetion to a ertain destination, to modify the bandwidth assigned to a onnetion, or to release a onnetion The atual establishment, maintenane, and release of ATM onnetions is performed by the signalling system For this purpose, the signalling system must interat with ATM layer entities in all systems along the route of the onnetion In the ase of an establishment or a request for additional bandwidth, negotiation is done between the requesting protool entity, the destination protool entity, and the signalling system (on behalf of the ATM network) The signalling system onfirms the suess or failure of the request to the onnetion management funtion No negotiation is needed for the release of a onnetion Basially, a onnetionless servie over ATM an be provided in two different ways: depending on the appliation either an end-to-end protool or a node-by-node protool an be used The end-to-end protool orresponds to the ase with onnetionless protool entities only in the end systems In order to transfer a paket from one end system to another end system, use should be made of an ATM onnetion between these end systems This way of providing a onnetionless servie is referred to as the indiret method within the ATM reommendations of the International Teleommuniation Union [4] The nodeby-node protool orresponds to the presene of one or more intermediate onnetionless protool entities alled Connetionless Servers (CLSs) If an end system has to transfer a paket, it has to use an ATM onnetion to a CLS In this CLS, the paket is routed to a onnetion with the appropriate next node, and so on Finally, in the destination node, the paket is delivered to the reeiving end-system C ATM swith E ATM swith D ATM swith F Figure 1 Use of ATM onnetions to provide a onnetionless servie end-system D servie user The ITU refers to this way of providing a onnetionless servie as the diret method Figure 1 illustrates the different types of ATM onnetions that an be used in both methods A line in the figure represents a set of two onnetions, one for eah diretion Here, the indiret method is used between endsystems A and B End-system B also employs the diret method, and therefore has a onnetion with CLS B Endsystems C and D are onneted to CLS A, and both CLSs are also onneted In order to maintain the overlay network of onnetions properly, the onnetion management funtion must analyse the need for the transfer of pakets Here a problem arises: in priniple, the protool entity does not have advane knowledge about the destination and the required bandwidth of the ATM onnetions it will need This is beause of the onnetionless nature of the ommuniation, where only individual pakets are transferred, whih are routed independently through the network In pratie, a ertain orrelation, in the destination as well as in the arrival time, an be expeted due to the behaviour of (appliation) protools in the higher layers An appliation for whih a single paket is transferred is likely to generate more data, so that more pakets to the same destination will follow within a limited period of time This expeted orrelation an be exploited by the onnetion management funtion, ie it an be used in the establishment, modifiation and release of ATM onnetions For example, the expetation that there is a large amount of traffi between two LANs whih have to be interonneted an lead to the establishment of a permanent onnetion between these LANs The new mehanism we present in the following setion exploits this orrelation to a large extent 22 Candidate mehanisms Several mehanisms for setting up ATM onnetions for the transfer of pakets an be envisaged The objetive of any suh mehanism is to minimize the load that is put on the ATM network for maintaining onnetions Bandwidth that is reserved for these onnetions annot be used for other (onnetion-oriented) appliations Furthermore, the load on the signalling system for establishing, modifying, 54

4 Connetion management mehanism and releasing onnetions must not be too high Too many signalling operations ould result in an overloaded signalling system, and hene in a degraded signalling performane, eg in a high onnetion setup delay Finally, the delay experiened by pakets in an end-system or CLS must be aeptable Let us now disuss the andidate onnetion management mehanisms First, we desribe two onventional mehanisms 221 Connetion per paket (CpP) A onnetion, neessary to transfer a paket, an be established as soon as the paket arrives at a protool entity The onnetion is released again immediately after the transfer of the paket This mehanism does not exploit the expeted orrelation between pakets All neessary knowledge, ie destination and amount of data to be transported, is known at the moment the onnetion is established Note that a paket will in general be sent in several ATM- SDUs (ells) A pratial extension might therefore be to transfer subsequent pakets with the same destination, arriving during the transmission time of the urrent paket, over the same onnetion 222 Permanent onnetion (PC) Alternatively to the previous mehanism, a protool entity an maintain one or more permanent onnetions to various possible destination entities A paket arriving in this protool entity is transferred on one of these onnetions Note that the entity must maintain onnetions to all entities to whih it must be able to transfer pakets diretly Other entities may be reahed in several steps, via intermediate entities The exat speifiation of the required bandwidth is a problem for this mehanism In priniple, the CL protool entity has no advane knowledge about the arrival times and the lengths of the pakets It an only use information about subsription, and statistis to predit the required bandwidth Optionally, the onnetion management mehanism may modify the bandwidth of the onnetion during the lifetime of a onnetion [5] 223 On-demand onnetion with delayed release (OCDR) In this mehanism a onnetion will be established if a paket has arrived, and no onnetion to the proper destination protool entity is available The onnetion will not be released immediately after transferring the paket It an be used for onseutive pakets to the same destination as well The onnetion will be released if it has not been used for a ertain period of time, the holding time This mehanism tries to exploit the expeted lustering of arrivals of onseutive pakets for the same destination entity It assumes that the expeted time until the next arrival is longer after the holding time has expired than immediately after a departure Thus, it an redue the time a onnetion has to be maintained, ompared to the PC mehanism, and at the same time, redue the mean delay experiened by pakets, ompared to the CpP mehanism The CpP and PC mehanisms are speial ases of this one: the OCDR mehanism with zero holding time is equivalent to the CpP mehanism with the mentioned extension, and the PC mehanism is equivalent to the OCDR mehanism with an infinite holding time 224 Disussion The desribed mehanisms an be applied between two end-systems, between an end-system and a CLS, as well as between two CLSs The hoie for a ertain mehanism and the aompanying parameters (eg holding time) an be made for eah pair of (soure and destination) protool entities individually The hoie will depend on the expeted arrival times and paket lengths of the traffi from the soure protool entity to the destination protool entity These depend heavily on the expeted appliations Furthermore, the arrival times depend also on the extent to whih traffi from different appliations or end-systems an be multiplexed onto one onnetion If the diret method of providing a onnetionless servie is used, pakets to different end-systems an be multiplexed, eg on the onnetion to the first CLS, or between CLSs For the indiret method, pakets to different end-systems must use different onnetions, sine end-to-end onnetions are used in this ase Note that an end-system an for instane be a router or bridge to a LAN, so that it does already multiplex the traffi of different LAN stations The amount of traffi that must be transferred over onnetions between CLSs an be expeted to be so high that permanent onnetions must be maintained between these CLSs Between whih CLSs onnetions must be established, and how muh bandwidth must be assigned to the onnetions is a dimensioning problem that is similar to dimensioning problems that an be found in traditional data networks [6, 7] In this paper, we fous on mehanisms for the establishment and release of fixed-bandwidth onnetions The proposed OCDR mehanism is a mehanism that an provide for this If the diret method of providing a onnetionless servie is employed, it an be used to onnet end-systems to an Aess CLS It an be applied on the onnetion from end-system to CLS and from CLS to end-system independently If the indiret method is employed, ie if end-systems need to be onneted diretly, and no CLSs are used, the OCDR mehanism an be used to ontrol the onnetion between end-systems 23 Performane riteria The three onnetion management mehanisms (CpP, PC, and OCDR) differ in a number of ways In the following setions, we will investigate what the performane differenes are The following performane measures are important for the evaluation of the mehanisms Average delay The average delay is the mean time elapsing from the arrival of a paket in the buffer of a CLS or endsystem until the departure of (the last ell of) the paket from the buffer This delay onsists of the time a The terms soure and destination do not neessarily refer to endsystems They must be interpreted relative to the onnetion, and an as suh refer to intermediate systems, ie CLSs 55

5 G Heijenk and B R Haverkort paket spends in a buffer plus the time neessary for transmission of the onseutive ells of a paket on an outgoing onnetion It an inlude the onnetion setup delay, if no onnetion is readily available Average reserved bandwidth The average reserved bandwidth is the long-term average bandwidth reserved on a onnetion between a soure/destination pair of CLSs Periods of time in whih no onnetion is available are taken into aount in this average, and are onsidered as periods during whih the reserved bandwidth is zero The average reserved bandwidth will be strongly related to the osts of the servie Average number of onnetion setups per seond The average number of onnetion setups per seond is the long-term average number of times a onnetion is established per seond This measure is an indiation for the load on the signalling system, whih is also a ost fator for the provision of a onnetionless servie Note that the number of requests for onnetion release equals the number of requests for onnetion establishment 3 Modelling and analysis In the previous setion, we have presented OCDR as one of the andidate mehanisms for the onnetion management funtion The purpose of this mehanism was to redue the reserved bandwidth, ompared to the PC mehanism, while reduing the delay, ompared to the CpP mehanism The purpose of this setion is to investigate this effet quantitatively Therefore, we model and analyse the mehanism to obtain results for the performane measures identified in setion 23 The modelling and analysis an also be used for the CpP and PC mehanisms, sine these are limiting ases of the OCDR mehanism We model the behaviour of the OCDR mehanism as far as a single pair of soure and destination nodes is onerned A onnetion between these entities may or may not exist The onnetion is established if a paket arrives in the soure node for the destination node under onsideration After finishing the transfer of a paket, the onnetion is released if no new paket for the destination arrives before the holding time expires We refrain from modelling the detailed behaviour at ell level, sine the paket level behaviour is dominant at the time-sale that is relevant for the onnetion management mehanisms Furthermore, the models in this setion are ontinuous-time models, sine the slotted nature of ATM is not relevant at the onsidered time-sale Table 1 gives an overview of the model parameters for later referene They will be introdued later on In setion 31, we first disuss the workload of the model, ie the stream of pakets arriving at the soure node that are destined for the destination node Two different traffi types are defined: Poisson traffi and bursty traffi In setion 32, we present the modelling and analysis of the OCDR mehanism assuming Poisson traffi Next, in setion 33, we present the modelling and analysis assuming bursty traffi Finally, in setion 34, we present Table 1 Model parameters 1/ Average paket interarrival time 1/ Average paket transmission time 1/r Average onnetion holding time 1/ Average onnetion setup time l Average paket length 1/ Average burst time 1/ Average interburst time n m Number of Erlang stages for the holding time Number of Erlang stages for the onnetion setup time an extension to the latter model, whih is based on more realisti assumptions for the holding time of the OCDR mehanism and the time needed to establish a onnetion 31 Workload We adopt two different workload haraterizations for the analysis of the OCDR mehanism The first one, Poisson traffi, results in an analytially tratable performane model It gives insight into the behaviour of the protool The seond one, bursty traffi, gives a more realisti haraterization of the expeted traffi, and therefore more aurate performane measures 311 Poisson traffi The onseutive interarrival times of pakets are assumed to be independent, and exponentially distributed with mean 1/ The operation of the OCDR mehanism is based on the assumption that there will be a orrelation in arrival times of subsequently transferred pakets Poisson traffi does not have this property Therefore, the advantages of the mehanism will not be revealed 312 Bursty traffi Here, we assume that pakets arrive in bursts In [8], it is shown that suh a bursty traffi soure, whih is modelled by a Train Model, provides a realisti desription of the traffi on a loal omputer network (Ethernet) In [9], it is shown that a Markov Modulated Poisson Proess (MMPP), whih an also be used to desribe the bursty traffi soure is very well suited to represent orrelations between subsequent arrivals We use the simplest MMPP possible, also known as the interrupted Poisson proess (IPP), for our evaluations It should be noted that with IPPs soures with extreme burstiness an be desribed [10] Let the stohasti proess {A(t), t > 0, A(t) {0,1}} desribe the arrival mode at time t IfA(t) = 1 the arrival proess is said to be in a burst, and pakets arrive at rate, ie with exponentially distributed interarrival times with mean 1/ If A(t) = 0, the arrival proess is said to be in an interburst period and no pakets arrive We assume the burst time and the interburst time to be exponentially distributed, with mean 1/ and 1/ respetively (see figure 2) It an easily be seen that the long-term mean of A(t), denoted E[A] an be expressed as E[A] = 1/ 1/ + 1/ = + (1) 56

6 Connetion management mehanism interburst time burst time distribution of this CTMC as P(i,j): burst interarrival time arrival P(i,j) = lim t {N(t) = i V(t)=j} (2) Figure 2 Behaviour of an interrupted Poisson proess The following system of balane equations an be obtained (equations (3) (6)): 0,0 1,0 i,0 i+1,0 ( + )P (i, 0) = P (i 1, 0) for i>1, (3) r by equating the flow into state (i, 0) to the flow out of the same state; 0,1 1,1 i,1 i+1,1 Figure 3 CTMC for OCDR under Poisson traffi P (i, 1) = (P (i 1, 1) + P(i 1,0)) for i>1, (4) by equating the flow aross the boundary between the states with N(t)>iand the states with N(t)<i; E[A] an be interpreted as the fration of time the arrival proess is in burst mode This implies that the long-term mean arrival rate is given by E[A] 313 Paket length distribution We assume the (appliation) paket length to be exponentially distributed The mean paket length is l bits Note that in a real situation, the paket length will have a disrete distribution However, the exponential distribution an be seen as a ontinuous time equivalent of the geometri distribution 32 OCDR under Poisson traffi Let us now present a model for the OCDR mehanism with Poisson arrivals The onerned model is a Continuous Time Markov Chain (CTMC) We are able to derive a losed-form solution for the stationary state probabilities of this model [11] When a paket arrives, and no onnetion exists, the onnetion management funtion invokes the signalling system in order to establish a onnetion Setting up a onnetion is assumed to take an exponentially distributed time with mean 1/ When the establishment of the onnetion has been onfirmed to the onnetion management funtion, the transmission of pakets an be started We assume that transmission of pakets takes plae at a rate of l bits per seond, ie every paket transmission takes an exponentially distributed time with mean 1/ seonds After all pakets have been sent, the onnetion will be released when the system is empty for an exponentially distributed holding time with mean 1/r Let the stohasti proess {N(t), t > 0, N(t) N} denote the number of pakets in the system at time t, ie the number of pakets in the soure entity, destined for the destination entity under onsideration Furthermore, let the stohasti proess {V(t), t > 0, V (t) {0, 1}}indiate whether or not a onnetion is available at time t Then, the proess {N(t), V (t)} is a CTMC In figure 3 the resulting state transition diagram is depited It should be noted that N(t) and V(t) are not independent We are interested in the steady state behaviour of this CTMC Let us denote the steady state probability rp(0,1) = P (0, 0), (5) by equating the flow into state (0, 0) to the flow out of this state; and (P (i, 0) + P(i,1)) = 1, (6) i=0 for normalization We will relate all stationary probabilities to P(0,0), and use the following notation: ρ =, (7) and σ = + (8) Notie that ρ is the long-term fration the server is busy, ie it is the utilization, expressing the inoming amount of work per unit of time Consequently, it must hold for stability: ρ<1 From equation (3) it follows diretly that, for all i, P(i,0) = σ i P(0,0) (9) From equation (4) and equation (9), we have for the states (i, j) with j = 1 P(i,1) = ρ(p(i 1,1) + σ i 1 P(0,0)) (10) Repeatedly substituting equation (10) into itself yields: P(i,1) = ρ i P(0,1)+ i ρ k σ i k P(0,0)) (11) k=1 Using equation (5), this results, for all i, in P(i,1) = ( r ρi + i ρ k σ )P(0,0) i k (12) k=1 We now realize that the long-term fration of time a server is busy must be equal to the utilization ρ, ie the following equality holds: P {server atually serving} = P(i,1) = ρ (13) i=1 57

7 G Heijenk and B R Haverkort By substituting equations (13), (5) and (9) in the normalization equation (6), we obtain: ( P(0,0) i=0 σ i + r ) +ρ = 1 (14) Rewriting the geometri series yields ( 1 P(0,0) 1 σ + ) +ρ = 1, (15) r whih, after substituting equation (8), results in the following expression for P(0,0): 1/ P(0,0) = (1 ρ) 1/ + 1/r + 1/ (16) Substitution of this expression in equations (9) and (12) leads to a losed-form solution for the steady state probabilities Having dedued this losed-form solution we an derive expressions for a number of performane measures For some of the measures, we make use of the Cauhy produt rule [12], aording to whih: i=0 k=0 i ( ) ρ k (i k)σ i k = iσ i ρ k i=0 i=0 for ρ,σ < 1 (17) The following measures have been derived for the evaluation of the OCDR mehanism: average number of pakets in the system, E[N]: E[N] = i(p(i,0) + P(i,1)) i=0 = ρ 1 ρ + ( ) + average delay, E[T ], using Little s law: 1/ 1/ + 1/r + 1/ ; (18) E[T ] = 1 E[N] = 1 ( ( ) ) ρ + 1 ρ + 1/ ; 1/ + 1/r + 1/ (19) the fration of time a onnetion is available, E[V ], using equation (13): E[V ] = P(i,1) i=0 1/r = ρ +(1 ρ) 1/ + 1/r + 1/ ; (20) average reserved bandwidth, E[B], in bit/s: E[B] = le[v ] ( ) 1/r = l ρ + (1 ρ) ; (21) 1/ + 1/r + 1/ average number of onnetion setups per seond, E[S]: E[S] = P(i,0) i=1 1 = (1 ρ) 1/ + 1/r + 1/ (22) In equation (18) and equation (19), the first term equals the M/M/1 result for the average number of pakets in the system and the average delay respetively In equation (20), the first term (ρ) orresponds to the fration of time that a onnetion is used for transmission The seond term expresses the time that an existing onnetion is idle 33 OCDR under IPP traffi In order to model the behaviour of the OCDR mehanism subjet to an IPP as desribed in setion 31, the CTMC of the previous setion needs to be extended The system is now modelled by the proess {N(t), V (t), A(t)}, whih is again a CTMC Reall that A(t) denotes the state of the arrival proess at time t The impliit assumption made at this point is that A(t) is independent of N(t) and V(t) As suh, reent investigations about the selfsimilarity of (Ethernet) traffi and the resulting existene of dependenies between these stohasti proesses, are not taken into aount (see also [13]) By making the above assumption, we will arrive at Markovian models whih an be analysed with known tehniques A more detailed investigation of the above mentioned dependenies (their existene in this ase, and their influene) goes beyond the sope of the urrent paper We are again interested in the steady-state behaviour of the CTMC We now define the steady state probabilities, P(i,j,k), as follows: P(i,j,k) = lim t P {N(t) = i V(t)=j A(t) = k} (23) The state transition diagram of this CTMC is depited in figure 4 It is similar to the one for Poisson arrivals (figure 3) The state spae is dupliated, to inorporate the state of the arrival proess The front plane of states (the states labelled (i, j, 1)) represents the situation where the arrival proess is in a burst The transitions between the states are idential to those for Poisson arrivals The bak plane of states (the states labelled (i, j, 0)) represents the situation where the arrival proess is in an interburst period It is idential to the front plane exept for the transitions with rate, whih have been removed to represent the absene of arrivals The system transits from the front plane to the bak plane and bak with rates and, respetively For the analysis of this CTMC, we use the matrix geometri solution method developed by Neuts ([2]; see also [3], [14], and [15]) Therefore, the repetitive struture of the CTMC is utilized Let the states be ordered lexiographially ((0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), et) Then the generator matrix Q of the CTMC 58

8 Connetion management mehanism 0,0,0 1,0,0 i,0,0 i+1,0,0 0,0,1 1,0,1 i,0,1 i+1,0,1 r r 0,1,0 1,1,0 i,1,0 i+1,1,0 0,1,1 1,1,1 i,1,1 i+1,1,1 Figure 4 CTMC for OCDR under IPP traffi an be written in a blok-triagonal form as follows: B 00 B B 00 A 1 A A Q = 2 A 1 A A 2 A 1 A 0, (24) A 2 A 1 where all onstituent elements of Q are 4 4 matries, and 0 is a matrix with all zeros Q has a tridiagonal form, similar to the generator matrix of an M/M/1 queue However, here the onstituent elements are matries themselves, unlike in the M/M/1 model, where the onstituent elements are salars Aording to the matrix geometri solution tehnique, the stationary state probability distribution of the Markov proess an be easily obtained after numerially solving a matrix quadrati equation with A 0, A 1 and A 2 as oeffiients for the repeating behaviour, and a system of linear equations for the boundary behaviour For this purpose, we have used the software tool Xmgm [3] Let us now give the onstituent A- and B-matries of the generator matrix Q The set of states of the CTMC for whih the number of ustomers in the system equals i is alled level i Transitions between levels orrespond to the arrival or departure of a paket Transitions within a level orrespond to a hange in the state of the arrival proess, or a hange in the presene of the onnetion The matrix A 0 gives the transition rates between states, given that the system goes from level i to level i+1; its entries orrespond to paket arrivals It is defined as follows: A 0 = (25) A 2 desribes the transitions between states, while the system transits from level i to level i 1 its entries orrespond to paket departures (servies): A 2 = (26) Finally, A 1 desribes the transitions within a level, ie the hange of arrival proess mode and the setup of a onnetion, and on the diagonal, the zero-row-sum ompensation for the omplete generator matrix: A 1 ( + ) 0 0 = ( + + ) ( + ) 0 0 ( + + ) (27) The B-matries define the boundary transitions, ie the transitions to and/or from level 0 It turns out that for the CTMC desribed in this subsetion, the transitions to and from level 0 are the same as for other levels, ie B 01 = A 0, (28) and B 10 = A 2 (29) The transitions within level 0 desribe the release of a onnetion as well as hanges in the arrival proess mode, and on the diagonal, the usual zero-row-sum ompensation: B 00 = () 0 0 ( + ) 0 0 r 0 ( + r) 0 r ( + + r) (30) We defer the expliit desription of the measures of interest to the end of the next setion 34 OCDR with Erlang holding and onnetion setup times Up to now, we have modelled the holding time of the OCDR mehanism as an exponentially distributed time In the real system, this time will probably be determined by a fixed timer value, and hene be a deterministi one In order to model this holding time more aurately, and to hek our previous model, we will now model it as an Erlang n distribution, ie a distribution with n idential exponentially distributed stages It is known that the squared oeffiient of variation of suh a distribution approahes 0 if n, ie the Erlang distribution approahes a deterministi distribution [16] Similarly, we model the onnetion setup time with an Erlang m distribution, sine this time an also be assumed to have a lower variane than that of an exponential distribution Let us now enhane the model that we have developed for OCDR under IPP arrivals, in order to ope with the Erlang distributions for the holding time and the onnetion setup time The system is again modelled with the CTMC (N(t), V (t), A(t)), but now the stohasti proess V(t) is defined to represent the stage of the Erlang distribution for the holding time or onnetion setup time as follows For an empty system, ie N(t) = 0, {V(t), t > 0, V (t) {0,1,,n}} defines the stage of the holding time, whih is initially n Assuming the system stays empty, V(t) dereases with a rate nr If V(t) = 0 the 59

9 G Heijenk and B R Haverkort 0,0,0 1,0,0 i,0,0 i+1,0,0 nr 0,0,1 1,0,1 i,0,1 i+1,0,1 nr 0,1,0 1,1,0 i,1,0 i+1,1,0 nr 0,1,1 1,1,1 i,1,1 i+1,1,1 nr nr 0,n,0 1,m,0 i,m,0 i+1,m,0 nr 0,n,1 1,m,1 i,m,1 i+1,m,1 Figure 5 CTMC of OCDR with Erlang holding and onnetion setup times onnetion has been released, otherwise it is still available If the system beomes non-empty, ie N(t) > 0, before the onnetion has been released, the timer is reset, ie V(t) starts at n again the next time the system beomes empty For a non-empty system, ie N(t) > 0, {V(t), t > 0, V (t) {0,1,,m}} defines the stage of the onnetion setup time, whih is initially 0 Now, V(t) inreases with a rate If V(t) = m, the onnetion has been set up, otherwise it is not yet available The stationary state probability distribution of the CTMC is still defined as in equation (23), however, note that the domain of V(t) has hanged The state transition diagram of the CTMC is shown in figure 5 Its generator matrix Q has the same shape as the one in equation (24) However, the onstituent matries are defined differently A 0, A 1 and A 2 are square matries of size 2(m + 1) 2(m + 1), with similar semantis as before They are defined by equations A 0 = , (31) A 2 =, (32) and equation (33) shown in figure 6 B 01 is a matrix of size (n + 1) (m + 1): B 01 = (34) B 10 is a matrix of size (m + 1) (n + 1), whih is defined similarly to A 2 : B 10 = (35) Finally, B 00 is a matrix of size (m + 1) (m + 1), whih is defined as equation (36) shown in figure 7 Now that we have defined the onstituent matries of the generator matrix, we are able to solve for the stationary state probabilities of the CTMC, again using Xmgm From the obtained probabilities we an derive a number of performane measures: average number of pakets in the system, E[N]: m E[N] = i (P (i, j, 0) + P(i,j,1)); (37) i=1 j=0 average delay, E[T ], using Little s law: E[T ] = 1 E[N]; (38) E[A] the fration of time a onnetion is available, E[V ]: E[V ] = + (P(i,m,0)+P(i,m,10)) i=1 n (P(0,j,0)+P(0,j,1)); (39) j=1 average reserved bandwidth, E[B]: E[B] = le[v ]; (40) average number of onnetion setups per seond, E[S], by reognizing that the establishment of a onnetion needs to be performed eah time a paket arrives while the system is empty and no onnetion is available: E[S] = P (0, 0, 1) (41) Note that this model is equivalent to the CTMC of the previous setion for n = 1 and m = 1 The expressions for the performane measures (equations (37) to (41)) apply also to that model if n and m are set to 1 4 Evaluation Using the models desribed in subsetions 32 34, we will evaluate the performane of the OCDR mehanism This will be done by omparing it to the CpP and PC mehanisms, with respet to the performane measures identified in setion 23 In setion 41 we will present the values for the model parameters that have been used Then, in setion 42, we 60

10 A 1 = ( + ) ( + + ) ( + ) ( + + ) 0 0 B 00 = Connetion management mehanism (33) ( + ) ( + + ) Figure 6 Equation (33) ( + ) nr 0 ( + nr) nr ( + + nr) ( + nr) ( + + nr) Figure 7 Equation (36) (36) evaluate the performane of OCDR with a workload of Poisson traffi In setion 43, we evaluate OCDR with a workload of bursty (IPP) traffi In setion 44, we do the same, now assuming that the holding time and onnetion setup time have an Erlang distribution In the last three subsetions, we ontinue to assume bursty traffi and Erlang holding and onnetion setup times In setion 45, we analyse the behaviour of OCDR under varying holding time, in order to determine the optimal value for this ontrol parameter Finally, in setions 46 and 47, we respetively evaluate the performane of OCDR under varying load and burst length 41 Parameter values The values for the model parameters, given in this subsetion, are the default parameters used in the experiments Unless stated differently, these values are used They are summarized in table 2 Values may be different for different models Furthermore, sometimes no default value is assumed for a parameter ( ), or a parameter is not appliable (na) to a model beause it is not defined for that model The workload parameters for the IPP traffi are based on measurements in [8], taking into aount the high-speed harater of future appliations The average interburst time has been taken to be 25 seonds, ie = 004 The average burst time is 1 seond, ie = 1 The average number of arrivals per burst is 100, ie = 100 This is an order of magnitude higher than the value measured in [8], to reflet the expeted inrease in traffi intensity in the future For Poisson traffi, we only have to define the parameter of the exponential distribution of the interarrival time, In order to have the average number of arrivals per seond equivalent for IPP traffi and Poisson traffi, we have adopted = 100 E[A] = 100 /( +) = 100/ for Poisson traffi The average paket length, l, is assumed to be 10 kbit for both Poisson and bursty traffi E [B] (bits/s) large r = 0 (PC) r = 1 r = 10 r = 100 small E [T] (s) Figure 8 Average reserved bandwidth plotted against average delay (Poisson traffi) The average of the onnetion setup time, 1/, is assumed to be 100 ms, ie = 10 The default value for the rate at whih pakets are served if a onnetion is available has been hosen suh that the utilization during a burst is 08, ie = 125 Furthermore, experiments (see, eg figure 11) have suggested a holding time of 01 seond (r = 10) as an optimal value In the model with Erlang holding and onnetion setup times, the number of stages for the holding time has been taken to be 30 The onnetion setup time has an Erlang 5 distribution 42 Poisson traffi The expeted gain of the OCDR mehanism is that it laims less resoures from the ATM network than a PC mehanism, beause an outgoing onnetion is released during periods in whih no traffi arrives The average reserved bandwidth (E[B]) is a good measure to express this laim, ie it is an indiation of the osts of the use of the ATM network A disadvantage of the use of OCDR, ompared to PC, is that some pakets will experiene a higher delay, beause a onnetion must be 61

11 G Heijenk and B R Haverkort Table 2 Default values for model parameters Poisson traffi IPP traffi IPP traffi (exponential holding and (Erlang holding and onnetion setup times) onnetion setup times) (s 1 ) 100/ (s 1 ) 125 r (s 1 ) 10 (s 1 ) l (bit) (s 1 ) na 1 1 (s 1 ) na n na na 30 m na na 5 established expliitly before pakets an be transferred However, by serving pakets at a high rate (), if the onnetion is available, ie by requesting a high bandwidth for the onnetion, the average delay (E[T ]) an be kept aeptable The problem we are interested in is the following Given a ertain load (), and a ertain required average delay (E[T ]), what holding time (1/r) and what onnetion bandwidth () should be hosen to ahieve a minimal average reserved bandwidth (E[B]), ie minimal osts? In order to deal with this problem, we investigate how the obtained average reserved bandwidth relates to the obtained average delay, for different values of the onnetion bandwidth () In figure 8, we display both measures for varying, ie the urves that are drawn in the figure are parametri urves is inreasing from right to left along the urves As an be observed, the average delay dereases, and the average reserved bandwidth inreases with inreasing Curves are depited for different holding times 1/r Note that an infinite holding time (r = 0) orresponds to the PC mehanism, and that a zero holding time (approahed by r = ) orresponds to the CpP mehanism Let us disuss the harateristis of the graph in more detail All the urves with r>0onverge to some vertial asymptote, ie they show an asymptoti behaviour to a limiting value of E[T ] This is the limiting behaviour for, for whih equation (19) redues to 1/ + 1/) lim E[T ] = (1/) (1/ + 1/r + 1/) (42) In this expression, 1/ is the expeted time a ustomer has to wait if no onnetion is available upon arrival The remaining fator an be onsidered as the probability that no onnetion is available for an arriving ustomer We diretly see that this implies that for the CpP mehanism (r ) the limiting value of the average delay equals 1/, the onnetion setup time, sine all pakets will find no onnetion available For r = 0, the limit of E[T ]is 0 for, whih implies that for the PC mehanism every delay demand an be guaranteed However, due to the fat that is finite beause of the finite apaity of an ATM link, this limit of 0 is never reahed From equation (19) and (21), it an be derived that all the urves ross at E[T ] = 1/ Note that in this point the values for differ for the various urves Only the average reserved bandwidth, whih is the produt of, the average paket length (l), and the fration of time a onnetion is available (equation (20)) is onstant We see from the urves that the optimal value for r for some average delay is either r = 0 when the required average delay is smaller than 01 s, or r when the required average delay is larger than 01 s Sine the required average delay an be expeted to be in the order of 001 s, the PC mehanism (r = 0) is the only suitable mehanism, if traffi arrives aording to a Poisson proess Conluding we an state that the OCDR mehanism is not advantageous if pakets arrive to a CLS or end-system aording to a Poisson arrival proess Either the CpP or PC mehanism needs the least bandwidth to fulfil some delay demand This is due to the fat that with the Poisson proess arrivals do not luster 43 IPP traffi with exponential holding and onnetion setup times Results for the OCDR model assuming arrivals aording to an IPP and exponential holding and onnetion setup times are obtained using Xmgm with equations (25 30) as input Analogously to the previous subsetion, we plot the average reserved bandwidth versus the average delay (see figure 9) Curves have been drawn for the same values of r as in setion 42 Again, the parameter varies along the urves Contrary to the OCDR mehanism under Poisson traffi, the mehanism is now advantageous if pakets arrive to the CLS or end-system aording to an IPP Depending on the required average node delay, one of the values for r yields the lowest average reserved bandwidth It an roughly be said that a PC mehanism (r = 0) is the optimal solution for a required average delay of less than 001, and a CpP mehanism (r ) for a required average delay of more than 01 For average delay requirements between 001 and 01, other values for r are optimal 44 IPP traffi with Erlang holding and onnetion setup times In the graph of figure 9, we have still assumed that the holding time and the onnetion setup time are 62

12 Connetion management mehanism large r = 0 (PC) small 020 =200 E [B] (bits/s) 10 6 r = 1 r = 10 r = 100 E [S] (s -1 ) =150 = = E [T] (s) Figure 9 Average reserved bandwidth plotted against average delay (IPP, n =1,m= 1) /r (s) Figure 11 Average number of onnetion setups per seond for varying holding time E [B] (bits/s) r = 0 (PC) large small r = 10 r = 1 r = E [T] (s) Figure 10 Average reserved bandwidth plotted against average delay (IPP, n = 30, m = 5) exponentially distributed In a real system this will definitely not be true The holding time will be a deterministi one, and the onnetion setup time will also have a lower variane than an exponential distribution In order to model the real system more aurately, we assume that both the holding time and the onnetion setup time are distributed aording to an Erlang distribution To reflet the fat that the holding time will be deterministi, we model it with an Erlang 30 distribution, ie n = 30 in figure 5 For the onnetion setup time, we assume an Erlang 5 distribution, ie m = 5 In [14], it is shown that the results from the model are not very sensitive to the exat values of m and n, in the range of system parameters that has been used The mentioned distributions are used throughout the rest of the setion Let us again give the same type of graph as in the previous two subsetions Figure 10 gives urves of the average reserved bandwidth versus the average delay The most onspiuous differene between this graph and the previous one appears in the urve for r = 10 This urve has shifted to the left, ie a lower average delay an be provided with the same average reserved bandwidth As a result, OCDR, with a holding time of 01 s, is the most optimal mehanism for a wide range of delay requirements Why is a holding time of 01 s the most optimal one? If the holding time is (an order of magnitude) lower (eg r = 100), it is frequently shorter than the paket interarrival time during a burst Consequently, the onnetion is released for a very short time, beause the next paket of the burst will arrive shortly, and hene, only little bandwidth is saved at the ost of an inreased average delay If the holding time is (an order of magnitude) higher (eg r = 1), it is of the same order as the length of the entire burst, and bandwidth is wasted beause the onnetion is not released fast enough, ie not even between bursts Now, the differene between the results for exponential and Erlang holding time also beome lear If the holding time were taken from an exponential distribution instead of an Erlang (or deterministi) one, its atual value would often be far different from the mean of the distribution As a result, the OCDR mehanism would perform worse, despite the optimal mean holding time For a required average delay in the range between 0007 s and 006 s, the OCDR mehanism with a holding time of 01 s an fulfil the delay requirements at the lowest ost, ie with the lowest average reserved bandwidth This range an expeted to be the operational region of the mehanism Compared to the PC mehanism a signifiant redution of the average reserved bandwidth an be obtained, ie up to 95% (for E[T ] = 006) 45 Optimal holding time From now on, we assume IPP traffi and Erlang holding and onnetion setup times In order to obtain more insight in the optimal value for the onnetion holding time, we do an experiment where we vary this parameter, keeping the other parameters onstant (see table 2 for the default values) We perform the experiment for different values of, so that the utilization during a burst is between 05 ( = 200) and 10 ( = 100) Here we only show a graph for the average number of onnetion setups per seond (E[S]) as a funtion of the holding time (1/r), in figure 11 The influene of the holding time on the behaviour of the OCDR mehanism beomes very lear from the graph If the holding time is larger than 01 seond, the rate at whih new onnetions are established is very lose to the rate at whih new bursts start, ie one per 26 seonds This indiates the desired behaviour of OCDR: the onnetion is released between bursts, and held during bursts If the 63

13 G Heijenk and B R Haverkort r = 0 (PC) 03 E [T] (s) 02 E [B] (bits/s) 10 5 r = r = 10 r = 0 (PC) (s -1 ) Figure 12 Average delay under varying load (s -1 ) Figure 13 Average reserved bandwidth under varying load holding time beomes muh larger, the average number of onnetion setups per seond dereases, beause the onnetion will no longer be released between bursts The mehanism behaves like a PC mehanism For small holding times (smaller than 001 seond), the rate at whih onnetions are set up depends on the utilization during a burst In general, this rate is high, beause the onnetion is often released during a burst, espeially for low utilizations (ie high ) If the utilization is high ( = 100), this is not the ase beause the system will only rarely beome empty during a burst In the range of 1/r between 001 s and 01 s, the average number of onnetion setups per seond sharply dereases with inreasing holding time Conluding, we an state that for the given parameters, a holding time of 01 s is optimal for the proper operation of the OCDR mehanism 46 Behaviour under varying load In setion 44, we have ompared the OCDR mehanism to the PC and CpP mehanisms It turned out that the OCDR mehanism an redue the average reserved bandwidth by up to 95%, depending on the required average delay Of ourse, the obtained results depend on the parameters of the arrival proess In order to obtain insight in this dependeny, the effet of the load and the burst length on the results will be investigated below In order to vary the load on the onnetionless protool, is varied Inreasing means that the interarrival time in a burst dereases, and the number of arrivals per burst inreases is kept onstant at 125, whih implies that the bandwidth reserved for a onnetion is 125 Mbit/s (l), if the onnetion is established We give urves for three values of r A holding time of 01 s (r = 10) turned out to be most optimal for OCDR in the previous subsetion For omparison, we also display urves for r = 0 (PC mehanism) and r = (approximating the CpP mehanism) Graphs are given for the average delay, E[T ] (figure 12), the average reserved bandwidth, E[B] (figure 13), and the average number of onnetion setups per seond, E[S] (figure 14) From figure 12, it an be observed that for the same, the average delay for OCDR is about twie the delay for PC if is between 50 and 125 This range orresponds to a utilization of the onnetion during a burst between 04 and 10 Reall that is the arrival rate during a burst The CpP mehanism has a muh higher delay here, beause the onnetion is often released during a burst The same an be observed for the OCDR mehanism if dereases below 50 If the load approahes zero, the mean delay for both the CpP and the OCDR mehanism approahes the sum of the paket transmission time and the onnetion setup time (1/ + 1/ = 0108 s), beause a new onnetion needs to be set up for every paket In that ase, the mean delay of the PC mehanism beomes the paket transmission time (1/ = 0008 s), beause a paket will not experiene waiting time any more For > the average delays for OCDR and CpP onverge, beause the system will no longer beome empty during a burst, due to temporary overload Consequently, the CpP mehanism will not release the onnetion during the burst From = 150, the differene in delay between the mehanisms will be onstant with inreasing load The delay of both mehanisms will inrease almost linearly If the load is so high that the system annot transmit the exess traffi of a burst in an interburst period any more, the delay will inrease more rapidly with inreasing load In figure 13, it an be seen that the average reserved bandwidth of the OCDR mehanism is almost onstant for > 50 This indiates that the OCDR mehanism is stable with these parameters: the onnetion is available during bursts, and released between bursts, regardless of the traffi intensity during the burst Of ourse, the average reserved bandwidth in the ase of a permanent onnetion (PC) is onstant at 125 Mbit/s For the CpP mehanism, only the bandwidth that is really used is reserved, ie E[B] = E[A]l = /2600 Mbit/s (see equation (1)) The mehanism does not waste any bandwidth As a result, the differene between the urve for CpP and the urves for the other mehanisms an be interpreted as the bandwidth that is wasted by the onerned mehanisms If is larger than, ie >125, OCDR wastes only very little bandwidth The PC mehanism on the other hand uses about 20 times as muh For high loads, a small differene between the average reserved bandwidth of the CpP and OCDR mehanisms remains, beause of the bandwidth reserved during the holding times, after eah burst When the system beomes 64