Call Blocking Probabilities of Multirate Elastic Traffic under the Threshold and Bandwidth Reservation Policies

Transcription

1 all Blocing Probabilities of Multirate Elastic raffic under the hreshold and Bandwidth Reservation Policies I D Moscholios, M D Logothetis, A Boucouvalas and V G Vassilais Dept of Informatics & elecommunications, University of Peloponnese, ripolis, Greece WL, Dept of Electrical & omputer Engineering, University of Patras, Patras, Greece Dept of Electronic Engineering, University of Surrey, GU2 7XH Guildford, U s: idm@uopgr, mlogo@upatrasgr, acb@uopgr, vvasilais@surreyacu Abstract We propose a multirate teletraffic loss model of a lin that accommodates different service-classes of elastic calls alls arrive in the lin according to a Poisson process, can tolerate bandwidth compression and have an exponentially distributed service time When bandwidth compression occurs, the service time of new and in-service calls increases All calls compete for the available lin bandwidth under the combination of the hreshold (H) and the Bandwidth Reservation (BR) policies he H policy can provide different QoS among service-classes by limiting the number of calls of a service-class up to a predefined threshold, which can be different for each service-class he BR policy reserves part of the available lin bandwidth to benefit calls of high bandwidth requirements he analysis of the proposed model is based on approximate but recursive formulas, whereby we determine call blocing probabilities and lin utilization he accuracy of the proposed formulas is verified through simulation and found to be very satisfactory I INRODUION Multirate elastic traffic refers to in-service calls of different service-classes which have the ability to compress/expand their bandwidth and simultaneously increase/decrease their service time, during their lifetime in a system Assuming that the system behaves as a loss system (ie, calls are not allowed to wait in order to be serviced) and that the call arrival process is Poisson then the calculation of various performance measures such as all Blocing Probabilities (BP) and system s utilization can be based on the classical Erlang Multirate Loss Model (EMLM) []-[2] he latter has led to numerous loss models proposed for the call-level analysis of wired (eg, [3]-[4]), wireless (eg, [5]-[23]) and optical networs (eg, [24]-[26]) In the EMLM, a lin of certain capacity accommodates calls of different service-classes New calls compete for the available lin bandwidth according to the complete sharing policy (ie, calls compete for all bandwidth resources) and have fixed bandwidth requirements and generally distributed service time [] he term fixed means that in-service calls do not compress their bandwidth during their lifetime in the system A new call is bloced and lost if its required bandwidth is not available he steady state probabilities in the EMLM have a Product Form Solution (PFS) which leads to an accurate BP calculation [], [2] In [5], the EMLM is extended to include the case of elastic traffic We name this model Elastic EMLM (E-EMLM) In the E-EMLM, instead of rejecting immediately a bloced call, the lin may accept this call by compressing its bandwidth and the bandwidth of all in-service calls Bandwidth compression is permitted down to a minimum bandwidth which can be different for each service-class Elastic calls increase their service time so that the product bandwidth by service time remains constant When a call with compressed bandwidth leaves the system, then the remaining calls expand their bandwidth in proportion to their initial bandwidth requirement all blocing occurs when the value of the minimum bandwidth requirement is still higher than the available bandwidth In this paper, we consider a lin that accommodates Poisson arriving calls of elastic service-classes and modify the admission mechanism to include the hreshold (H) and the Bandwidth Reservation (BR) policies We name the proposed model E-EMLM/H- BR In the H policy, the number n of in-service calls of service-class should not exceed a pre-defined threshold, after the acceptance of a new service-class call Otherwise, the call is bloced and lost he H policy is significant since it analyzes a multirate access tree networ which accommodates different serviceclasses [27] and may differentiate service-classes in terms of BP or revenue rates by a proper threshold selection (see eg, [28], [29]) he BR policy is used to reserve bandwidth to benefit calls of high bandwidth requirements and is mainly used when BP equalization is required among calls of different service-classes he fact that the BR policy has been extensively applied in the literature (eg, [6]-[7], [4], [2], [3]-[33]) evinces its importance in call admission control o model the proposed E-EMLM/H-BR, we use the Marov chain method However, due to the existence of the compression/expansion mechanism and the BR policy, the reversibility of the Marov chains is destroyed, and the steady state probabilities in the proposed E-EMLM/H-BR cannot be determined via a PFS herefore, we resort to approximate Marov chains which provide recursive formulas for the efficient determination of the lin occupancy distribution and, consequently, BP and lin utilization he accuracy of the proposed formulas is verified through simulation and found to be very satisfactory his paper is organized as follows In Section II, we present the E-EMLM/H and prove formulas for the

2 calculation of the various performance measures In Section III, we extend the E-EMLM/H to include the BR policy In Section IV, we provide numerical results whereby the E-EMLM/H and the E-EMLM/H-BR are compared to existing models and evaluated via simulation We conclude in Section V II HE ELASI EMLM/H (E-EMLM/H) A he system model onsider a lin of capacity bandwidth units (bu) that accommodates elastic service-classes alls of service-class (,, ) follow a Poisson process with arrival rate and request b bu Bandwidth compression is introduced in the system by allowing the occupied lin bandwidth j to virtually exceed up to bu, ie, j 0,,, Let n ( n,, n ) be the vector of all in-service calls and b ( b,, b ) the vector of pea-bandwidth requirements, then j nb he decision to accept a new service-class call in the system is based on the following constraints: a) he number of in-service calls of service-class, n, together with the new call, should not exceed a threshold n, ie, n n Otherwise the call is bloced his constraint expresses the H policy b) If constraint (a) is met then: b) if jb, the call is accepted in the system with b bu and remains in the system for an exponentially distributed service time with mean b2) if jb the call is accepted by compressing its b together with the bandwidth of all in-service calls of all service-classes he compressed bandwidth of the new service-class call is: b rb b ( j+b ) () r r( n) ( nb b) ( jb) he product service time by bandwidth per call is ept constant by changing the mean value of the service time of the new service-class call to μ ( j b) μ he compressed bandwidth of all in-service calls becomes bi bi ( jb) for i,, When all calls have compressed their bandwidth, then j Note that the minimum bandwidth that a call tolerates is: b r b b (2),min min rmin / is the min proportion of the required pea-bandwidth and is common for all service-classes A new service-class call is bloced if jb When an in-service call, with compressed bandwidth b i departs from the system then the remaining calls expand their bandwidth to b in proportion to their b i, as follows: " i " bi min bi, bi bb i nb (3) B he analytical model he existence of the bandwidth compression mechanism destroys reversibility in the E-EMLM/H and therefore the steady state probabilities have no PFS o circumvent this problem, we use state-dependent factors ( n ), which lead to a reversible Marov chain:, when nb and n in Ω (4) - ( n) x( n ), when nb and n in Ω x( n) n: 0 nb, n n,,,, n ( n,, n,, n), - n ( n,, n,, n ) and x() n nbx ( n), when nb, nin Ω (5) o prove a recursive formula for the lin occupancy distribution, G(j), we initially consider the global balance equation for state n, expressed as rate into state n = rate out of state n : P n n n P n P( n) n( n) P( n ) ( ) ( ) ( ) ( ) n ( n,, n,, n) and P( n), P( n), P( n) are the probability distributions of states nn,, n, respectively Assume now, the existence of Local Balance (LB) between adjacent states hen the following LB equations can be extracted, for,, and n : P( n ) n ( n) P( n ) (6) P( n) ( n ) ( n ) P( n ) (7) Based on the assumption of LB, P( n) can be determined by: n a P( n) G x( n ) (8) n! a / is the offered traffic-load (in erl) of n a service-class and G G( ) x( n ) nω n! Since j is the occupied lin bandwidth, G( j) is defined as: G( j) P( n), Ω n Ω : nb j (9) nω j j onsider now two sets: () 0 j and (2) j For set (), we have the EMLM/H and G( j ) s are given by the following formula [30]: G( j) a b G( jb ) ( jb ), for j,, (0) j

3 x Pr j x n n () ( ): [, ] In () the fact that n n implies that j nb When j, we substitute (4) in (6) to have: ax( n) P( n ) nx( n ) P( n ) (2) Multiplying both sides of (2) by b and summing over we obtain: x( n ) a b P( n ) P( n ) n b x( n ) (3) Equation (3), due to (5) is written as: P( n) abp( n ) (4) Summing both sides of (4) over Ω j n Ω : nb j and based on (9), we obtain: G( j) ab P( n) (5) n j Since n n then P( n ) G( jb) Pr xjb, nn n Ω j hus, (5) can be written as: Gj () ab Gj ( b) ( jb), forj,, (6) ( x) is given by () Equations (0), (6) result in the following approximate but recursive formula for the calculation of G(j) s in the E-EMLM/H: Gj () ab Gj ( b) ( jb), forj,, (7) min( j, ) Having determined G( j ) s we calculate the BP of service-class, B, and the lin utilization, U, as: B G G( j) G ( j) b jb jnb - - (8) (9) U jggj ( ) GGj ( ) j j G G( j) is the normalization constant j0 In (7) and (8) the nowledge of ( j) is required Since ( j ) >0 when j nb,, b, we consider two subsets: ) nb jand 2) j b For the first subset, let a system of capacity F bnb that accommodates all service-classes but service-class For this system, we define r ( j) as: r ( j) ab r ( jb) ( jb), for j,, F (20) i i i i i j i i Based on r ( j ) s, we compute ( j) via the formula: n a n! ( j) r ( j n b ) (2) For the second subset, ( j) can be determined by: a ( j) x( ) n a n ni i n! n i ni! i (22) n : nb nb i i j, jb i, i III HE ELASI EMLM/H-BR (E-EMLM/H-BR) onsider again a lin of capacity bu that accommodates elastic service-classes of Poisson arriving calls A new service-class (=,,) call has a pea-bandwidth requirement of b bu and a BR parameter t that expresses the reserved bu used to benefit calls of all other service-classes except If j b t and n + n then the call is accepted in the lin and remains for an exponentially distributed service time with mean Otherwise the call is bloced and lost o determine G(j) s in the E-EMLM/H-BR we propose the following approximate but recursive formula:, for j 0 Gj ( ) ad ( jb) Gj ( b) ( jb), forj,, min( j, ) 0, otherwise (23) b for jt D( j-b)= (24) 0 for j t A characteristic of the BR policy is that it ensures BP equalization among different service-classes by a proper selection of the BR parameters If, for example, BP equalization is required between calls of two service-classes with b = and b 2 =0 bu, respectively, then t = 9 bu and t 2 = 0 bu so that b + t = b 2 + t 2 he application of the BR policy in the E-EMLM/H- BR is based on the assumption that the number of service-class calls is negligible in states j > - t and is incorporated in (23) by the variable D (j-b ) given in (24) he states j > - t belong to the so-called reservation space Note that the population of calls of service-class in the reservation space may not be negligible In [6], [0] a complex procedure is implemented that taes into account this population in the EMLM and Engset multirate state-dependent loss models, respectively However, this procedure may not always increase the accuracy of the BP results compared to simulation [0] Similarly to the E-EMLM/H, we determine the BP of service-class, B, based on two groups of states: i) those where the available lin bandwidth is less than b + t bu when the new call arrives in the system; this happens when b t j and ii) those where the available lin bandwidth is enough to accept the new call, ie j b t but n n ; the latter implies

4 that j nb, or nb jb t hus, the values of B are calculated by: B G G( j) G ( j) b t j b t j nb (25) G G( j) is the normalization constant j0 As far as U is concerned, it can be determined by (9) In (23) and (25) the nowledge of (j) is required for nb jb t We consider again two subsets: ) nb jand 2) j b t For subset (), we can use (20), (2) where F b t nb, while for subset (2) we use (22) where x(n) is given by (5) and n : nb nb i i j, jb t i i If = and both the H and the BR policies are considered, then calls are not allowed to compress their bandwidth; in this case, the proposed E-EMLM/H-BR coincides with the EMLM/H-BR of [34] he values of G(j) s and BP are given by (26), (27), respectively:, for j 0 G( j) ad ( jb) G( jb) ( jb), for j,, j 0, otherwise B G G( j) G ( j) (26) b t (27) jb t jnb b for jt D( j-b)= 0 for j t he lin utilization can be calculated by: - jg G j j (28) U ( ) (29) Finally, if = and we do not consider the H and the BR policies, then the proposed E-EMLM/H-BR coincides with the classical EMLM of [], [2] In that case, the lin occupancy distribution is determined by the well-nown aufman-roberts recursion:, for j 0 G( j) abg( jb), for j,, (30) j 0, otherwise he BP of service-class is given by [], [2]: B G G j jb ( ) (3) while the lin utilization can be determined by (29) IV NUMERIAL EXAMPLES - EVALUAION In this section, we present an application example of the proposed E-EMLM/H-BR and the model of [34] (EMLM/H-BR) hrough the proposed model we obtain analytical BP and lin utilization results, and compare them with the corresponding simulation results, in order to reveal the accuracy of the proposed model he simulation model is based on the bandwidth compression/expansion mechanism described by r( n ) s Simulation results are mean values of 7 runs Each run is based on the generation of four million calls o account for a warm-up period, the blocing events of the first 5% of these generated calls are not considered in the results Due to the fact that reliability ranges are very small, they are not presented in the figures that follow he simulation language used is Simscript III [35] As an application example, we consider a lin of capacity = 70 bu and three values of : ) 70 bu, 2) 75 bu with r min 70 / 75 and 3) 80 bu with r min 70 / 80 he lin accommodates three service-classes, with the following characteristics: st service class : a 50 erl, b 2, n 25, t 7 nd 2 service class : a2 5 erl, b2 5, n2, t2 4 rd 3 service class : a3 0 erl, b3 9, n3 6, t3 0 In the x-axis of all figures, traffic loads α, α 2 and α 3 increase in steps of, 05 and 025 erl, respectively So, Point refers to ( a, a2, a3) (50,5,0) while Point 7 is ( a, a2, a3) (0, 45,25) In Figs -3, we consider the proposed E-EMLM/H- BR and present the analytical and simulation BP results of the three service-classes, respectively, for all values of For comparison, we present the corresponding analytical results of the EMLM/H-BR (when 70 ) According to Figs -3, we deduce that: (i) the results obtained by the proposed formulas are very close to the simulation results (ii) he bandwidth compression mechanism reduces BP as expected (higher reduction is achieved for =80 bu) (iii) he analytical BP results obtained by the existing EMLM/H-BR fail to approximate the simulation BP results of the E-EMLM/H-BR (iv) he application of the BR policy in the E- EMLM/H-BR results in the BP increase of the st and 2 nd service-classes and the BP decrease of the 3 rd service-class his behavior is expected since the BR parameters are chosen to favor the 3 rd service-class In Fig 4, we present the lin utilization results (in bu) Again, the analytical results are very close to simulation, while the existing EMLM/H-BR fails to approximate the results obtained by the proposed model V ONLUSION In this paper we propose a multirate loss model where Poisson arriving calls compete for the available lin bandwidth under the H and the BR policies alls are of elastic type, ie, they can tolerate bandwidth compression while in-service he analysis of the proposed model leads to approximate but recursive formulas for the

5 calculation of the steady-state probabilities and consequently BP and lin utilization Simulation results verify the accuracy of the proposed model Figure 3 BP 3 rd service-class Figure BP st service-class Figure 2 BP 2 nd service-class Figure 4 Utilization

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