Congestion Probabilities in a Batched Poisson Multirate. Loss Model Supporting Elastic and Adaptive Traffic

Transcription

1 Ann. Telecommun. manuscript No. (will be inserted by the editor) Congestion Probabilities in a Batched Poisson Multirate Loss Model Supporting Elastic and Adaptive Traffic Ioannis Moscholios John Vardaas Michael Logothetis Anthony Boucouvalas Received: February 20 / Accepted: date Abstract The ever increasing demand of elastic and adaptive services, where inservice calls can tolerate bandwidth compression/expansion, together with the Moscholios ID Boucouvalas AC Dept. of Telecommunications Science and Technology, University of Peloponnese, 22 00, Tripolis, Greece idm@uop.gr Boucouvalas AC acb@uop.gr Vardaas JS Logothetis MD Dept. of Electrical & Computer Engineering, University of Patras, Patras, Greece jvardaas@upatras.gr Logothetis MD mlogo@upatras.gr

2 2 Ioannis Moscholios et al. bursty nature of traffic, necessitate a proper teletraffic loss model which can contribute to the call-level performance evaluation of modern communication networs. In this paper, we propose a multirate loss model that supports elastic and adaptive traffic, under the assumption that calls arrive in a single lin according to a Batched Poisson process (a more bursty process than the Poisson process, where calls arrive in batches). We assume a general batch size distribution and the partial batch blocing discipline, whereby one or more calls of a new batch are bloced and lost, depending on the available bandwidth of the lin. The proposed model does not have a product form solution, and therefore we propose approximate but recursive formulas for the efficient calculation of time and call congestion probabilities, lin utilization, average number of calls in the system and average bandwidth allocated to calls. The consistency and the accuracy of the model are verified through simulation, and found to be quite satisfactory. Keywords Batched Poisson Process Elastic Adaptive Traffic Recursive Formula Time Call Congestion Marov Chain Introduction The call-level performance evaluation of wired (e.g. []-[8]), wireless (e.g. [9]-[4]) and optical networs (e.g. [5]-[9]) is usually based on the classical Erlang Multirate Loss Model (EMLM), where a single lin is modelled as a loss system that accommodates Poisson arriving calls of different service-classes. Calls have fixed bandwidth requirements and compete for the available lin bandwidth under the complete sharing policy (all calls compete for all available bandwidth resources). Call blocing occurs when the required bandwidth of a new call is higher than the

3 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 3 available lin bandwidth. The fact that the EMLM steady-state probabilities have a Product Form Solution (PFS) leads to an accurate calculation of Call Blocing Probabilities (CBP). In [20], [2], an accurate recursive formula, nown as Kaufman-Roberts recursion, has been proposed for the efficient calculation of the EMLM macro-state probabilities (lin occupancy distribution) and consequently CBP. Through the reduced load approximation method, the applicability of the EMLM and the Kaufman-Roberts recursion is broadened in loss networs [22]-[23]. In contemporary communication networs, the existence of elastic and adaptive service-classes together with the bursty nature of traffic weaens the applicability of the EMLM. In this paper, we propose a new teletraffic loss model to include these three features. To this end, initially we focus on the wor of [24],[25] where the notion of elastic and adaptive traffic is incorporated into the EMLM. Calls that can reduce their bandwidth, while simultaneously increasing their service time, compose the so-called elastic traffic (e.g. file transfer). Calls that can tolerate bandwidth compression, but their service time cannot be altered, compose the so-called adaptive traffic (e.g. adaptive video). In [25], Poisson arriving calls have a pea bandwidth requirement and can tolerate bandwidth compression down to a minimum value. If the occupied lin bandwidth does not exceed the lin capacity, all in-service calls use their pea-bandwidth requirements. When the required pea-bandwidth of a new call is higher than the available lin bandwidth, the system accepts the call (contrary to the EMLM where this call is bloced and lost) by compressing its pea-bandwidth, as well as the bandwidth of all in-service calls. A new call is bloced and lost when, after the maximum possible bandwidth compression, the minimum bandwidth requirement of the call is still higher than the available lin bandwidth. The minimum bandwidth requirement of a call is

4 4 Ioannis Moscholios et al. determined as a proportion of the required pea-bandwidth of the call; this proportion is common to all service-classes. When an in-service call, whose bandwidth has been compressed, departs from the system, then the remaining in-service calls (of all service-classes) expand their bandwidth. Note that in the bandwidth compression/expansion procedure, only the service time of elastic calls is adjusted accordingly, so that the product service time by bandwidth remains constant. The bandwidth compression/expansion destroys reversibility in the Marov chain of the model and therefore no PFS exists. However, in [25] an approximation is proposed through a reversible Marov chain, which leads to a recursive calculation of lin occupancy distribution. For presentation purposes we name, herein, the model of [25], Extended EMLM (E-EMLM). Extensions of the E-EMLM in wired or wireless networs are found in [5], [], [26]-[29], under the assumption of random arrival process, except from [5], where a quasi-random arrival process is assumed (offered traffic-load comes from a finite number of sources) [30]. In this paper, we extend the E-EMLM by assuming that elastic/adaptive calls arrive in the lin according to a Batched Poisson process. In this way, we tae into account the bursty nature of traffic, a sine qua non traffic feature. The Batched Poisson process is very important, because it can be used to approximate arrival processes that are more peaed and bursty than the Poisson process [3]. Teletraffic loss models with a Batched Poisson process have been proposed for calls with fixed bandwidth requirements in [32]-[36]. The EMLM with Batched Poisson arrivals is examined in [32], under the hypothesis of partial batch blocing (part of the batch i.e. one or more calls, can be bloced due to lac of bandwidth) and geometric batch size distribution. The latter is highly used as a batch size distribution, since it is memoryless and a discrete equivalent of the exponential

5 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 5 distribution [37]. The same model but with a general batch size distribution is considered in [33]. We name the model of [33], Batched Poisson EMLM (BP-EMLM). In the proposed new model, we also consider the partial batch blocing discipline together with a general batch size distribution. In [34], the BP-EMLM under the bandwidth reservation policy is considered. This policy reserves a fraction of the available lin bandwidth to benefit service-classes with higher bandwidth per call requirements. In [35], an asymptotic analysis of the BP-EMLM is presented, regarding CBP when the offered traffic-load and lin capacity are commensurately large. In [36], the BP-EMLM is extended to include: a) a sequence of lins, b) a state-dependent batch arrival process, c) complete batch blocing, that is, the entire batch is bloced even if only one call of the batch is bloced, and d) two more bandwidth sharing policies, namely the upper limit and guaranteed minimum policies. A teletraffic loss model with a Batched Poisson process and elastic traffic only, under the bandwidth reservation policy, appears in [38]. In the proposed new model no PFS exists, and therefore we propose a reversible Marov chain which leads to an approximate recursive formula for the efficient calculation of the lin occupancy distribution. This solution eeps the model tractable and simplifies the determination of Time Congestion (TC) and Call Congestion (CC) probabilities, lin utilization, average number of calls in the system and average bandwidth allocated to calls. A comparison between simulation and analysis shows the consistency and the accuracy of the new model. Furthermore, we discuss the relationship of the proposed new model with that of [39] or [40]. In principle, the models of [39], [40] aim at sharing the capacity of a single lin among Poisson arriving elastic calls already accepted for service, according to a balanced fairness criterion, whereas our model focuses on call ad-

6 6 Ioannis Moscholios et al. mission. We show that the bandwidth compression/expansion mechanism in these models is the same. The rest of this paper is organized as follows: In Section 2, we present the proposed new multirate loss model. In Subsection 2. we describe the model, and in Subsection 2.2 we provide the analysis for the calculation of the lin occupancy distribution. In Section 3, we firstly show (Subsection 3.) the relationship of the proposed model with the BP-EMLM, the E-EMLM and the EMLM, as well as the balanced fairness model of [40], and secondly (Subsection 3.2) provide formulas for the calculation of performance measures for all models. In Section 4, we evaluate the proposed model by providing both analytical and simulation results. We conclude in Section 5. In Appendix A, we present a tutorial example of the new multirate loss model. In Appendix B, we present the notion of state-dependent multipliers whereby Marov chain reversibility is achieved and provide formulas for their calculation. In Appendix C, we prove the formulas for the determination of the average number of calls per service-class (elastic or adaptive) in the system. 2 The proposed model 2. Basic Description Consider a lin of capacity C bandwidth units (b.u.) that accommodates calls of K different service-classes. A service-class ( =,..., K) can be elastic or adaptive. Let K e and K a be the set of elastic and adaptive service-classes (K e +K a = K), respectively. Calls of each service-class arrive in the lin according to a Batched Poisson process with arrival rate λ, follow the partial batch blocing discipline and request b b.u. (pea-bandwidth requirement). To introduce bandwidth com-

7 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 7 pression, we permit the occupied lin bandwidth j to virtually exceed C up to a limit of T b.u. Suppose that an arriving batch of service-class contains only one call, while the system is in state j. Then, for call admission, we consider three cases:. If j+ b C, no bandwidth compression taes place and the new call is accepted in the system with its pea-bandwidth requirement for an exponentially distributed service time with mean µ (all in-service calls continue to have their pea-bandwidth requirement). 2. If j + b > T, the call is bloced and lost. 3. If T j + b > C, the call is accepted in the system by compressing its peabandwidth requirement, as well as the assigned bandwidth of all in-service calls (of all service-classes). All calls (in-service and new) share the capacity C in proportion to their pea-bandwidth requirement, while the lin operates at its full capacity C. This is in fact the processor sharing discipline [4]. When T j + b > C, the bandwidth b compressed of the newly accepted call of service-class, is given by: b compressed = rb = C j b () where r = C/j denotes the compression factor r, and j = j + b. Since j = K n b = nb, where n is the number of in-service calls of service-class = (in the steady state), n = (n,n 2,...,n K ) and b = (b, b 2,...,b K ), the values of r can be expressed by r r(n)=c /(nb+b ). The bandwidth of all in-service calls is also compressed by the same factor r(n) and becomes equal to b compressed i = C j b i for i =,...,K. After bandwidth compression we always have j = C, and all elastic When no bandwidth compression taes place, i.e. nb + b C, then r r(n) =.

8 8 Ioannis Moscholios et al. calls (of all elastic service-classes) increase their service time so that the product service time by bandwidth remains constant. Adaptive calls (of all adaptive service-classes) do not alter their service time. The minimum bandwidth that a call (either new or in-service) of service-class ( =,...,K ) can tolerate is: b compressed,min = r min b = C T b (2) where r min = C /T is the minimum proportion of the required pea-bandwidth and is common to all service-class calls. When an in-service call of service-class, with bandwidth b compressed, departs from the system, then the remaining in-service calls (of all service-classes) expand their bandwidth to b expanded i (i=,...,k ), in proportion to their pea-bandwidth b i, as follows: b expanded i = min b i, b compressed i + b i b K n b = compressed To illustrate the compression mechanism, consider the following example. Let C =3 b.u., T =5 b.u., K =2 service-classes, b = b.u., b 2 =2 b.u and µ =µ 2 = time unit. Furthermore, assume that the st service-class is elastic and the 2 nd service-class adaptive. We denote by n, n 2 the number of in-service calls of the st and 2 nd service-class, respectively. The total number of states n = (n, n 2 ) of the system is 2. Compression is applied to calls whenever T j > C (i.e. j =4, 5); after compression has been applied, we have j = C = 3. Assume now, that a new 2 nd service-class batch of two adaptive calls arrives, while the system is in state (n, n 2 ) = (2, 0), where j = 2 b.u. Since j +2 b 2 = 6 > T only one out of these two calls is accepted in the system, and j = j +b 2 = 4 < T. This is achieved by compressing the bandwidth of all calls (the new call and in-service calls). The (3)

9 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 9 new state of the system is now (n, n 2 )=(2, ). In this state, according to eq. (), calls compress their bandwidth to: b compressed =rb = 3 4 b =0.75, b compressed 2 =rb 2 = 3 4 b 2 =.5, so that j =n b compressed +n 2 b compressed 2 = =3=C. Similarly, the value of µ becomes µ r = 4 3 µ, so that the product service time by bandwidth remains constant. On the other hand the value of µ 2 does not alter. Assume now that a st service-class call departs from state (n, n 2 )=(2, ). Then, its assigned bandwidth b compressed = 0.75 is shared to the remaining calls according to eq. (3). Thus, in the new state (n, n 2 ) = (, ), the st service-class call expands its bandwidth to b expanded =b = b.u. and the 2 nd service-class call to b expanded 2 =b 2 =2 b.u. so that j=n b +n 2 b 2 =3 b.u. Furthermore, the service time of the elastic call becomes µ = time unit. In Appendix A, we present this tutorial example to facilitate the reader understanding the details of the proposed model. 2.2 Determination of lin occupancy distribution The system under consideration cannot be described by a reversible Marov chain due to the bandwidth compression/expansion mechanism. Therefore, the steadystate distribution P(n) does not have a PFS. In our analysis, we consider a reversible Marov chain that approximates the system and derive an approximate but recursive formula for the efficient calculation of the lin occupancy distribution, G(j ), j =0,,...,T. Let Ω be the state space of the system, Ω={n : 0 nb T }, and n +m, n m a next and a previous state of the system state n, respectively: n +m =(n, n 2,..., n, n + m, n +,...,n K ), m=,2,... n m =(n, n 2,..., n, n m, n +,...,n K ), m=,2,..., n.

10 0 Ioannis Moscholios et al. A classical way to determine G(j ) s recursively in multirate loss models is to assume that local balance exists between adjacent states of Ω [20]. According to the definition of local balance between adjacent states, the service rate going out from a state due to a call departure equals to the arrival rate going into this state due to a call arrival (rate-down = rate-up). To illustrate the notion of local balance in our model, we consider again the example of Subsection 2.. A small excerpt of the state transition diagram of the system is depicted in Fig.. From Fig. it is apparent that the aforementioned local balance definition does not hold due to the Batched Poisson process. The latter implies that one state can be reached by previous states due to an arriving batch which contains one or more calls. However, a form of local rate balance equation across certain levels that separate two adjacent states can be introduced as in [33]. This type of local rate balance was sufficient in [33] to guarantee a PFS for P(n) but not in the proposed model, because the bandwidth compression/expansion mechanism destroys the local rate balance equations, i.e. the corresponding Marov chain becomes irreversible. In order to construct a reversible Marov chain that approximates our model, in addition to the notion of local rate balance across certain levels, we use state multipliers for all states n Ω. By the means of the state multipliers we shall approximate the local rate balance equations. State multipliers have been used ([24]-[26]) to construct reversible Marov chains in order to approximate Marov chains that are irreversible. In order to introduce state multipliers and reversibility in the Marov chain, the first step is to define the levels across which local balance should hold. For a state vector n separates n = (n,n 2,..., n,...,n K ), with n 0, the level L n from n=(n, n 2,..., n,...,n K ). As an example, Fig. shows the

11 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic Fig. An excerpt of the state space Ω for a system with two Batched Poisson service-classes. level L 2 (0,), which separates n 2 = (0,) from n=(0, 2). The total upward rate across the level L n, due to arrivals of batches of service-class is given by: n R (up) (L n ) = P (n l )λ l=0 m=l+ B () m (4) where n l =(n, n 2,...,n, n l, n +,...,n K ), with n l 0, and B m () is the probability that a new batch of service-class contains m calls. To understand eq.(4), consider R (up) (L 2 (0,)) in Fig., where n 2 = (n, n 2 ) = (0, ): R (up) (L 2 (0,) ) = l=0 P(0, )λ 2 P (n l 2 )λ 2 m= m=l+ B (2) m + P(0, 0)λ 2 B (2) m R (up) L 2 (0,) = m=2 B (2) m The st term of the right hand side expresses the fact that if the system is in state (0,), then a new batch of service-class should contain at least one call in order to pass the level L 2 (0,), while the 2nd term expresses the fact that the new batch should contain at least two calls to pass the level L 2 (0,) from state (0,0).

12 2 Ioannis Moscholios et al. Since B () l = m=l+ B() m is the complementary batch size distribution for service-class, eq. (4) taes the following equivalent form: n R (up) (L n ) = P (n l () )λ B l (5) l=0 The downward rate across the level L n, due to a departure of an elastic call of service-class is: R (down) (L n ) = P (n)r(n)µ n (6) On the other hand, the downward rate across the level L n call of service-class is: for an adaptive R (down) (L n ) = P (n)µ n (7) To understand eq. (6), consider n 2 = (0, ) and n = (0, 2), and determine R (down) (L 2 (0,) ) as: R (down) (L 2 (0,) ) = µ 2P (0, 2) =.5µ 2 P (0, 2) The compression/expansion mechanism of calls destroys local balance across the level L n, that is: R (up) (L n ) R (down) (L n ) (8) or, distinguishing elastic and adaptive service-classes, n l=0 P (n l )λ m=l+ B () m P (n)r(n)µ n, K e (9) n l=0 P (n l )λ m=l+ B () m P (n)µ n, K a (0)

13 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 3 The second step towards a reversible Marov chain is to convert the inequalities (9) and (0) to equalities. This is done by introducing to them the state dependent multipliers φ (n), as follows: n l=0 n l=0 P (n l P (n l )λ )λ m=l+ m=l+ B () m = P (n)φ (n)µ n, K e () B () m = P (n)φ (n)µ n, K a (2) where φ (n) is defined as in [25], in order for the Marov chain to become reversible:, when nb C and n Ω φ (n) = x(n x(n ), when C < nb T and n Ω ) 0, otherwise (3) and, when nb C, n Ω ( x(n)= C n b x(n )+r(n) ) n b x(n ), when C <nb T, n Ω K e K a 0, otherwise (4) where r(n) = C/(nb). When C < nb T and n Ω the values of bandwidth of all in-service calls (of all service-classes) should be compressed by a factor φ (n) so that: n b compressed + n b compressed = C (5) K e K a To derive eq. (4), we eep the product service time by bandwidth of service-class calls (elastic or adaptive) in state n of the irreversible Marov chain equal to the corresponding product in the same state n of the reversible Marov chain. This means that: b r(n) µ r(n) = bcompressed µ φ (n) b compressed = b φ (n), K e (6)

14 4 Ioannis Moscholios et al. and b r(n) = bcompressed µ µ φ (n) b compressed = b φ (n)r(n), K a (7) Equation (4) results by substituting eq. (6), eq. (7) and eq. (3) into eq. (5). Having defined φ (n) and x(n) according to eq. (3) and eq. (4), respectively, it can be verified that eq. () and eq. (2) hold. In Appendix B, we further explain the notion of φ (n) and derive formulas for their calculation. Now, in order to prove a recursive formula for the calculation of G(j ) s, we consider two set of states: i) those where 0 j C, ii) those where C < j T (bandwidth compression occurs). When 0 j C, then φ (n)= and based on eq.() and eq. (2), it is proved that [33]: for j = 0 K j/b () G(j ) = j α b B l G(j lb ) = l= 0 for j < 0 for j =,..., C (8) where: α = λ / µ is the offered traffic-load (in erl) of service-class calls, j /b is the maximum integer not exceeding j/b, and B () l = m=l+ B m (). When C < j T, we multiply both sides of eq. () by b compressed and sum over K e to have: α b compressed K e n l=0 P (n l ) m=l+ B () m =P (n) n b compressed φ (n) (9) K e Based on eq.(3) and eq. (6) and since B () l = B m (), eq.(9) can be written m=l+ as: x(n) K e α b n l=0 P (n l ) B () l = P (n) x(n )n b (20) K e

15 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 5 We continue by multiplying both sides of eq.(2) by b compressed and sum over K a to have: α b compressed K a n l=0 P (n l ) m=l+ B () m = P (n) n b compressed φ (n) (2) K a Based on eq. (3) and eq. (7) and since r(n) = C /j, eq. (2) can be written as: x(n) C j K a α b n l=0 P (n l ) B () l = P (n) C j K a x(n )n b (22) Adding eq. (20) and eq. (22) we obtain: ( ) n x(n) α b P (n l ) () Bl + C n j K a α b P (n l ) () Bl = K e l=0 l=0 ( ) (23) P (n) x(n )n b + C j x(n )n b K e K a Due to eq. (4), eq. (23) can be written as: K e α b n l=0 P (n l ) () Bl + C j K a α b n l=0 P (n l ) () Bl = CP (n) (24) To introduce the lin occupancy distribution G(j ) in eq. (24), let Ω j ={n Ω:nb=j} be the state space where exactly j b.u. are occupied. Then, summing both sides of eq. (24) over Ω j we obtain: n α b P (n l n Ω j K e l=0 ) B () l + n Ω j C j since, by definition, n Ω j P (n) = G(j). n α b P (n l K a l=0 Interchanging the order of summations in eq. (25), we have: ) () Bl = CG(j) (25) n α b P (n l ) B K e n Ω j l=0 () l + C j α b K a n Ω j n l=0 P (n l ) () Bl =CG(j) (26) or K e α b j/b l= G(j lb ) B () l + C j K a α b j/b l= since n n Ω j l=0 P (n l ) = j/b G(j lb l= ). G(j lb ) () Bl = CG(j) (27)

16 6 Ioannis Moscholios et al. The combination of eqs. (8), (27) gives the approximate recursive formula of G(j ) s, when j T : G(j )= for j = 0 min(j,c) K e α b j/b l= j/b l G(j lb )+ j α b K a l= B () for j =,..., T B () l G(j lb ) (28) 0 for j < 0 In eq. (28), the batch size is generally distributed. If calls of service-class arrive in the system in batches of size s, where s is given by the geometric distribution with parameter β, i.e. Pr(s = x) = (-β )β x, with x, then since B () G(j)= l = β l, eq. (28) taes the form: for j = 0 min(j,c) K e α b j/b l= β l for j =,..., T 0 for j < 0 j/b G(j lb )+ j α b K a l= β l G(j lb ) (29) 3 Relationship of the proposed model with other multirate loss models and calculation of various performance measures 3. Relationship to other models a) If C = T, then no compression taes place and the BP-EMLM results ([33]). In this case, eq. (28) is identical to eq. (8). If the batch size is geometrically distributed, eq. (29) is identical to eq. (9) of [32]. b) If C = T, and B () m = for m = and B () = 0 for m >, the batch always contains one call and a Poisson process results. In that case, eq. (28) taes the form m

17 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 7 of the classical Kaufman-Roberts recursion used to accurately calculate G(j ) s in the EMLM ([20]-[2]): for j = 0 K G(j ) = j α b G(j b ) for j =,..., C = 0 otherwise (30) c) If C < T, and B () m = for m = and B () =0 for m > (Poisson process), elastic and adaptive calls can tolerate bandwidth compression/expansion. In that m case, eq. (28) taes the form of: for j = 0 G(j )= α min(j,c) b G(j b ) + j α b G(j b ) K e K a 0 otherwise used to calculate G(j ) s in the E-EMLM [25]. for j =,..., T (3) d) If C < T, and B () m = for m = and B () = 0 for m > (Poisson process), and we have only elastic service-classes in the system, then eq. (28) becomes: for j = 0 K G(j ) = α b G(j b ) for j =,..., T min(j,c) used to calculate G(j ) s in [24]. = 0 otherwise m (32) The proof of eq. (32) is based on φ (n) s as defined in eq. (3), while x(n) s are defined as follows [24]:, when nb C, n Ω K x(n) = n b x(n ), when C < nb T, n Ω C = 0, otherwise (33)

18 8 Ioannis Moscholios et al. e) Having shown the relationship of our model with that of [24], we now discuss the relationship of the model of [24] with that of [40]. In [40], a lin of capacity C b.u. accommodates calls of K service-classes of elastic type only. Arrivals follow a Poisson process. The lin capacity is shared among calls according to the balanced fairness criterion: When nb C, all calls use their pea-bandwidth requirement, while when nb > C, all calls share the capacity C in proportion to their peabandwidth requirement and the lin operates at its full capacity C. The main difference between the models of [24] and [40] lies on the fact that in [24] the notion of T >C allows for admission control, whereas there is no such parameter in [40]. The application of balanced fairness in multirate networs and its comparison with other classical bandwidth allocation policies, such as max-min fairness and proportional fairness, can be found in [42],[43]. In what follows, we show that the bandwidth allocated to calls when C < nb T is the same either by using the compression/expansion mechanism of [24] or the balanced fairness criterion of [40]. We do not present the case of nb C, since both models let calls use their pea-bandwidth requirements. According to [40], the balanced fair sharing φ (n) of all service-class calls in state n = (n, n 2,..., n,..., n K ) is given by: φ (n) = Φ(n e ), n Φ(n) > 0 (34) where Φ(n) is a balance function, defined by [40]: 0, if n < 0 for some Φ(n) = n!b, if n 0, nb C n K Φ(n e ), if n > C C = (35)

19 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 9 and n! = K = n!, b n = K = bn, while e is the unit line vector with in component and 0 elsewhere. According to eq. (35), when C < nb, eq. (34) is written as [40]: φ (n) = C Φ(n e ) K Φ(n e ) = (36) For presentation purposes let K = 2 service-classes. Then, assuming that in state n e calls of service-class use their pea-bandwidth requirement, the balanced fairness allocation gives: φ (n, n 2 ) = C (n )!n 2!b n b n 2 2 (n )!n 2!b n b n (n )!(n 2 )!b n bn 2 2 n = b C (n b + n 2 b 2 ) (37) Similarly: φ n 2(n, n 2 ) = 2 b 2 C (n b + n 2 b 2 ) (38) So, in state n = (n, n 2 ), where C <nb, the bandwidth allocated to a serviceclass call is: b compressed = φ (n, n 2 ) n b = C, =, 2 (39) (n b + n 2 b 2 ) Based on eq. (3) and eq. (33), the model of [24] gives: φ (n) = φ (n, n 2 ) = C n b + n 2 b 2, =, 2 (40) assuming that x ( ) n = (i.e. no compression in state n ). According to eq. (6), b compressed = b φ (n, n 2 ), and because of eq. (40), eq. (39) results; that is, the model of [24] results in the same values with those provided by balanced fairness.

20 20 Ioannis Moscholios et al. 3.2 Performance metrics for all models Having determined G(j ) s according to the individual formula of each model (i.e. eq. (8), eq. (28), eq. (30) and eq. (3)), we calculate the following performance measures: a) The TC probabilities of service-class, denoted as P b, which is the probability that at least T b + b.u are occupied: where: G = P b = T j=t b + T G(j ) is a normalization constant. j =0 G G(j) (4) TC probabilities are determined by the proportion of time the system is congested. An outside observer can measure this probability. In the BP-EMLM and the EMLM, we can use eq. (4) by replacing T with C. b) The CC probabilities of service-class, denoted as C b, which is the probability that a new service-class call is bloced and lost: where B () m C b = T G G(j) j=0 m= T j b + B () m (42) is the probability that a new batch of service-class contains m calls and (T j )/b is the maximum integer not exceeding (T j)/b. If the batch size is geometrically distributed, then since B β l, eq. (42) becomes: C b = () l = m=l+ B () m = T T j G b G(j)β (43) j=0 The proof of eq. (42) is similar to the proof of eq. (0) of [32] (which gives the CC probability in the BP-EMLM where the batch size is geometrically distributed) and therefore is omitted.

21 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 2 CC probabilities are determined by the proportion of arriving calls that find the system congested. An inside observer (i.e. an arriving call) can measure this probability. Since TC and CC probabilities coincide in the case of Poisson arrivals (PASTA property: Poisson Arrivals See Time Averages [30]), we can use eq. (4) for the calculation of CC probabilities in the case of the EMLM, E-EMLM. In the BP- EMLM, we can use eq. (42) or eq. (43) by replacing T with C. c) The lin utilization, denoted as U : C T U = jg G(j) + CG G(j) (44) j= j=c+ Equation (44) is valid for the proposed model and the E-EMLM. In the BP-EMLM and the EMLM, where T = C, the lin utilization is given by: C U = jg G(j) (45) j= d) The average number of service-class calls in the system, n : T n = y (j)g G(j ) (46) j = where y (j) is the average number of service-class calls in state j. The product of y (j )G(j) is given by the following eq. (47) in the case of elastic service-classes and by eq. (48) in the case of adaptive service-classes (for proof, see Appendix C): j /b y (j)g(j) = min(c,j) α b G(j xb ) (y (j xb ) + x) min(c,j) K a j i= K e i=, i j /b i α i b i x= j /b i α i b i x = x= G(j xb i )y (j xb i ) G(j xb i )y (j xb i ) m=x B (i) m m=x B (i) m + m=x B () m + (47)

22 22 Ioannis Moscholios et al. j /b y (j)g(j) = j α b K a j i=, i K e min(c,j) i= j /b i α i b i x= j /b i α i b i x= x= G(j xb ) (y (j xb ) + x) G(j xb i )y (j xb i ) m=x G(j xb i )y (j xb i ) B (i) m + m=x B (i) m m=x B () m + e) The average bandwidth allocated to service-class calls in the system, b : (48) j= b = T b r(j)y (j)g(j) T y (j)g(j) j= (49) where: r(j) = min (, C/j). The value of b = b if C = T, and b < b if C <T. 4 Numerical Results For evaluation, we present a realistic application example of the proposed model and provide both the analytical and simulation results of TC probabilities, CC probabilities and lin utilization. In addition, we provide the corresponding analytical results of the BP-EMLM, for comparison. This comparison is essential in order to show the consistency and the necessity of the proposed model. Regarding simulation results, they are obtained through SIMSCRIPT II.5 [44], as mean values of 7 runs with 95% confidence interval. Since the simulation results of the proposed model are very close to the analytical results, we choose to present only analytical results in figures, and simulation results together with the analytical in tables, for comparison. Consider a single lin of capacity C = 300 b.u. that accommodates four serviceclasses. The first three service-classes are of elastic type, while the fourth serviceclass is of adaptive type. Initially, calls require b = b.u., b 2 = 4 b.u., b 3 = 5

23 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 23 b.u. and b 4 =20 b.u., respectively. The batch size of all service-classes follows the geometric distribution with parameters β = 0.75, β 2 = 0.5, β 3 = 0.2 and β 4 = 0.2, respectively. The call holding time is exponentially distributed with the same mean values: µ = µ 2 = µ 3 = µ 4 =. The offered-traffic loads initially are: α = 4.0 erl, α 2 =5.0 erl, α 3 =3.0 erl and α 4 =.0 erl. In the horizontal axis of Figs. 2-6, we assume that the values of α 3, α 4 remain constant, while the values of α and α 2 increase in steps of 2.0 erl and.0 erl, respectively. Thus, in the horizontal axis of Figs. 2-6, Point corresponds to (α = 4.0 erl, α 2 = 5.0 erl, α 3 = 3.0 erl, α 4 =.0 erl), Point 2 corresponds to (α =6.0 erl, α 2 =6.0 erl, α 3 =3.0 erl, α 4 =.0 erl) and Point 8 corresponds to (α =28.0 erl, α 2 =2.0 erl, α 3 =3.0 erl and α 4 =.0 erl). We consider two different values of T : i) T =C =300 b.u., where no bandwidth compression taes place. In that case, the proposed model gives exactly the same results (for all performance measures) with the BP-EMLM. ii) T = 320 b.u., where bandwidth compression taes place with r min = C/T = 300/320. Table Simulation Results of the TC Probabilities of the Proposed Model for T = 320 b.u. (α, α 2 ) erl Service-class Service-class 2 Service-class 3 Service-class 4 (4.0, 5.0) ± ± ± ± (6.0, 6.0) ± ± ± ± (8.0, 7.0) ± ± ± ± (20.0, 8.0) ± ± ± ± (22.0, 9.0) ± ± ± ± (24.0, 0.0) ± ± ± ± (26.0,.0) ± ± ± ± (28.0, 2.0) ± ± ± ±0.0087

24 24 Ioannis Moscholios et al. In Figs. 2-5, we present the analytical results of the CC and TC probabilities, respectively, for both values of T. In all figures we observe that: a) CC and TC probabilities decrease as the value of T increases, since more calls are admitted due to the existence of the bandwidth compression mechanism and b) the CC probabilities are higher than the corresponding TC probabilities. The latter happens in the Batched Poisson process, because CC probabilities refer to the total number of bloced calls out of the total number of arriving calls, whereas TC probabilities refer to the total number of times the entire batch (of calls) is bloced (the system is full) out of the total number of arriving calls [34]. In Tables, 2, we show the corresponding simulation and analytical results for the TC probabilities, in the case of T = 320 b.u. In Tables 3, 4, we present the corresponding results for the CC probabilities. In Fig. 6, we show the analytical results of the lin utilization in b.u. for both values of T. Since the CC and TC probabilities of the proposed model are lower than those of the BP-EMLM, the proposed model achieves higher lin utilization. Table 2 Analytical Results of the TC Probabilities of the Proposed Model for T = 320 b.u. (α, α 2 ) erl Service-class Service-class 2 Service-class 3 Service-class 4 (4.0, 5.0) (6.0, 6.0) (8.0, 7.0) (20.0, 8.0) (22.0, 9.0) (24.0, 0.0) (26.0,.0) (28.0, 2.0)

25 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 25 Table 5 contains both simulation and analytical results of the lin utilization, when T = 320 b.u. In order to show that the proposed model behaves quite well even under high or extraordinary bandwidth compression, we consider the following values of T: 400, 500, 600 b.u. For these values of T, we comparatively present in Tables 6, 7 the analytical and simulation results of TC and CC probabilities, respectively, for each service-class, when the offered traffic-loads are: α = 28.0 erl, α 2 = 2.0 erl, α 3 =3.0 erl and α 4 =.0 erl. 5 Conclusion We propose a multirate loss model that supports elastic and adaptive traffic, under the assumption that calls arrive in a lin of fixed bandwidth capacity, according to a Batched Poisson process. The batch size is generally distributed, while calls are accepted or rejected by the system according to the partial batch blocing Table 3 Simulation Results of the CC Probabilities of the Proposed Model for T = 320 b.u. (α, α 2 ) erl Service-class Service-class 2 Service-class 3 Service-class 4 (4.0, 5.0) ± ± ± ± (6.0, 6.0) ± ± ± ± (8.0, 7.0) ± ± ± ± (20.0, 8.0) ± ± ± ± (22.0, 9.0) 0.023± ± ± ±0.002 (24.0, 0.0) ± ± ± ± (26.0,.0) ± ± ± ± (28.0, 2.0) ± ± ± ±0.0064

26 26 Ioannis Moscholios et al. discipline. The proposed model does not have a PFS and therefore we propose approximate but recursive formulas for the efficient calculation of the most significant performance measures such as time and call congestion probabilities, lin utilization, average number of calls in the system and average bandwidth allocated to calls. Simulation results verify the analytical results. Furthermore, we show the consistency of the proposed model and its relationship with other multirate loss models in the literature. A Tutorial example Consider again the example of Subsection 2.. A lin of capacity C = 3 b.u. and T = 5 b.u. accommodates calls of two service-classes. The st service-class is elastic and the 2 nd service-class is adaptive. Remind that b = b.u., b 2 = 2 b.u., while µ = µ 2 = time unit. Furthermore, we assume that the batch size of both service-classes follows the geometric distribution with parameters β = 0.2 and β 2 = 0.5, which corresponds to.25 and 2.0 mean number of call arrivals per batch, respectively. The system has 2 states n = (n, n 2 ), which are presented in Table 8, together with the occupied lin bandwidth j in each state, and the Table 4 Analytical Results of the CC Probabilities of the Proposed Model for T = 320 b.u. (α, α 2 ) erl Service-class Service-class 2 Service-class 3 Service-class 4 (4.0, 5.0) (6.0, 6.0) (8.0, 7.0) (20.0, 8.0) (22.0, 9.0) (24.0, 0.0) (26.0,.0) (28.0, 2.0)

27 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 27 state dependent factors r(n) = C/j, φ (n), φ 2 (n). The values of φ (n), φ 2 (n) are calculated through eq. (3) and eq. (4). The global balance equation for a state n = (n, n 2 ) is of the form: K=2 (R (up) (L n ) R (up) (L n )) = = K=2 = (R (down) (L n ) R (down) (L n )) (50) When we use the state dependent factor r(n) (irreversible Marov chain) in eq. (50), we get the following system of equations: State n =(0, 0) : P (0, ) + P (, 0) 2.0P (0, 0) =0 State n =(0, ) : 0.5P (0, 0) + 2.0P (0, 2) + P (, ) 3.0P (0, ) =0 State n =(0, 2) : 0.5P (0, 0) + P (0, ) + 0.6P (, 2) 3.0P (0, 2) =0 State n =(, 0) : 0.8P (0, 0) + P (, ) + 2.0P (2, 0) 3.0P (, 0) =0 State n =(, ) : 0.8P (0, ) + 0.5P (, 0) + 2.0P (, 2) +.5P (2, ) 4.0P (, ) =0 State n =(, 2) : P (0, 2) + 0.5P (, 0) + P (, ) 2.6P (, 2) =0 State n =(2, 0) : 0.6P (0, 0) + 0.8P (, 0) + 3.0P (3, 0) + P (2, ) 4.0P (2, 0) =0 State n =(2, ) : 0.6P (0, ) + 0.8P (, ) + P (2, 0) +.8P (3, ) 3.5P (2, ) =0 State n =(3, 0) : 0.032P (0, 0) + 0.6P (, 0) + 0.8P (2, 0) + P (3, ) + 3.0P (4, 0) 5.0P (3, 0) =0 State n =(3, ) : 0.04P (0, ) + 0.2P (, ) + P (2, ) + P (3, 0) 2.8P (3, ) =0 Table 5 Simulation and Analytical Results of the Lin Utilization of the Proposed Model for T = 320 b.u. (α, α 2 ) Simulation Results Analytical Results in erl in b.u. in b.u. (4.0, 5.0) ± (6.0, 6.0) 9.5 ± (8.0, 7.0) ± (20.0, 8.0) ± (22.0, 9.0) ± (24.0, 0.0) ± (26.0,.0) ± (28.0, 2.0) ±

28 28 Ioannis Moscholios et al. Fig. 2 CC and TC probabilities of the st service-class calls. State n =(4, 0) : P (0, 0)+0.032P (, 0)+0.6P (2, 0)+0.80P (3, 0)+3.0P (5, 0) 4.0P (4, 0) =0 State n =(5, 0) : 0.006P (0, 0)+0.008P (, 0)+0.04P (2, 0)+0.20P (3, 0)+P (4, 0) 3.0P (5, 0) =0 Its solution is: P (0, 0) = P (0, ) = 0.66 P (0, 2) = P (, 0) = P (, ) = P (, 2) = 0.05 P (2, 0) = P (2, ) = P (3, 0) = P (3, ) = P (4, 0) = P (5, 0) = 0.02 Based on the values of P (n) s, the exact values of G(j) s are: G(0) = 0.250, G() = P (, 0) = 0.334, G(2) = P (0, )+P (2, 0) = , G(3) = P (, )+P (3, 0) = 0.70, G(4) = P (0, 2)+P (2, )+P (4, 0)=0.923, G(5)=P (, 2)+P (3, )+P (5, 0)= Thus, the exact values of TC and CC probabilities are: P b =0.754, P b2 =0.3677, C b =0.2225, C b2 = When we use the state dependent factors φ (n), φ 2 (n) (reversible Marov chain) in eq. (50), we get the following system of equations:

29 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 29 Fig. 3 CC and TC probabilities of the 2nd service-class calls. State n =(0, 0): P (0, )+P (, 0) 2.0P (0, 0)=0 State n =(0, ): 0.5P (0, 0)+2.0P (0, 2)+P (, ) 3.0P (0, )=0 State n =(0, 2): 0.5P (0, 0)+P (0, ) P (, 2) 3.0P (0, 2)=0 State n =(, 0): 0.8P (0, 0)+P (, )+2.0P (2, 0) 3.0P (, 0)=0 State n =(, ): 0.8P (0, )+0.5P (, 0)+.7647P (, 2)+.7428P (2, ) 4.0P (, )=0 State n =(, 2): P (0, 2)+0.5P (, 0)+P (, ) P (, 2)=0 State n =(2, 0): 0.6P (0, 0)+0.8P (, 0)+3.0P (3, 0) P (2, ) 4.0P (2, 0)=0 State n =(2, ): 0.6P (0, )+0.8P (, )+P (2, 0) P (3, ) P (2, )=0 State n =(3, 0): 0.032P (0, 0)+0.6P (, 0)+0.8P (2, 0) P (3, )+3.0P (4, 0) 5.0P (3, 0)=0 State n =(3, ): 0.04P (0, )+0.2P (, )+P (2, )+P (3, 0) P (3, )=0 State n =(4, 0): P (0, 0)+0.032P (, 0)+0.6P (2, 0)+0.80P (3, 0)+3.0P (5, 0) 4.0P (4, 0)=0

30 30 Ioannis Moscholios et al. Fig. 4 CC and TC probabilities of the 3rd service-class calls. State n =(5, 0): 0.006P (0, 0)+0.008P (, 0)+0.04P (2, 0)+0.20P (3, 0)+P (4, 0) 3.0P (5, 0)=0 The solution of this system is: P (0, 0) = P (0, ) = P (0, 2) = P (, 0) = P (, ) = P (, 2) = P (2, 0) = P (2, ) = P (3, 0) = P (3, ) = P (4, 0) = P (5, 0) = 0.00 Based on the values of P (n) s, the approximate values of G(j) s are: G(0) = 0.277, G() = P (, 0) = 0.276, G(2) = P (0, )+P (2, 0) = , G(3) = P (, )+P (3, 0) = 0.632, G(4)=P (0, 2)+P (2, )+P (4, 0)=0.2044, and G(5)=P (, 2)+P (3, )+P (5, 0)= The same values of G(j) s can also be obtained through eq. (29). Thus, the approximate values of TC and CC probabilities (based on eq. (32) and eq. (34), respectively) are: P b = 0.729, P b2 = , C b = , C b2 = These results are very close to the

31 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 3 Fig. 5 CC and TC probabilities of the 4th service-class calls. corresponding exact values given above and show that the proposed formula can give quite good results even in the case of small examples. B Notion of state-dependent multipliers φ (n) and their calculation Consider a lin that accommodates calls of two service-classes. The first service-class is elastic and the second is adaptive. When nb C the bandwidth b and the service rate µ, =,2, of calls are not affected. In this case φ (n)=, =,2, where n=(n,n 2 ) and b=(b, b 2 ). When T nb > C, the bandwidth b and the service rate µ, of calls is decreased by φ (n), =,2, so that the lin operates at its full capacity: n b compressed + n 2 b compressed 2 = C (5)

32 32 Ioannis Moscholios et al. Fig. 6 Lin utilization (b.u.). Furthermore, from eq. (6) and eq. (7) we have: b compressed = b φ (n) (elastic service class) (52) b compressed 2 = b 2 φ 2 (n)r(n) (adaptive service class) (53) Based on eq. (52) and eq. (53), eq. (5) becomes: n b φ (n) + n 2 b 2 φ 2 (n)r(n) = C (54) Consider now an excerpt of the state transition diagram of our system that consists of four adjacent states as shown in Fig. 7. According to Kolmogorov s criterion [4], the system Marov chain (Fig. 7) becomes reversible if: Flow clocwise = Flow counter-clocwise

33 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 33 By applying this criterion to our Marov chain (Fig. 7), we have: λ φ 2 (n, n 2 )n 2 µ 2 φ (n, n 2 )n µ λ 2 = φ (n, n 2 )n µ φ 2 (n, n 2 )n 2 µ 2 λ λ 2 (55) In eq. (55) we notice that the Kolmogorov s criterion holds by a proper selection of the state-dependent multipliers φ (n), which affect the bandwidth and service rate requirements (not the arrival rates). Equation (55) is simplified to: φ 2 (n, n 2 )φ (n, n 2 ) = φ 2 (n, n 2 )φ (n, n 2 ) (56) Table 6 Simulation and Analytical Results of the TC Probabilities of the Proposed Model for T = 400, 500, 600 b.u. Service T =400 b.u. T =500 b.u. T =600 b.u. -class Simulation Analyt. Simulation Analyt. Simulation Analyt. st ± ± ± nd ± ± ± rd ± ± ± th ± ± ± Table 7 Simulation and Analytical Results of the CC Probabilities of the Proposed Model for T = 400, 500, 600 b.u. Service T =400 b.u. T =500 b.u. T =600 b.u. -class Simulation Analyt. Simulation Analyt. Simulation Analyt. st 0.05± ± ± nd ± ± ± rd ± ± ± th ± ± ±

34 34 Ioannis Moscholios et al. By choosing:, when nb C and n Ω φ (n) = x(n ) x(n, when C < nb T and n Ω ) 0, otherwise (57) eq. (56) holds and the Marov chain becomes reversible. Equation (54), based on eq. (57), is written as: x(n) = C [ n b x(n ) + r(n)n 2b 2 x(n 2 )] (58) where r(n) = C/(nb). Equation (58) is eq. (4) for two service-classes (elastic and adaptive). Table 8 State Space, Occupied Lin Bandwidth and State Dependent Factors of the Tutorial Example n n 2 j r(n) φ (n), φ 2 (n) φ (n), φ 2 (n)

35 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 35 Fig. 7 State transition diagram of four adjacent states. To determine φ (n) and φ 2 (n) we proceed as follows. Based on eq. (54) and eq. (56) we get: φ (n, n 2 ) = φ 2 (n, n 2 ) = C n b, if n and n 2 = 0 Cφ (n,n 2 ) n b φ (n,n 2 )+r(n,n 2 )n 2 b 2 φ 2 (n,n 2 ), if n and n 2 C r(n,n 2 )n 2 b 2, if n 2 and n = 0 Cφ 2 (n,n 2 ) n b φ (n,n 2 )+r(n,n 2 )n 2 b 2 φ 2 (n,n 2 ), if n and n 2 (59) For K service-classes, when n, K, we have: φ (n,..., n,..., n K ) = Cφ (n,...,n,n,n +,...,n K ) n i b i φ i (n,...,n i,n i,n i+,...,n Ke )+r(n ) i Ke i Ka n i b i φ i (n,...,n i,n i,n i+,...,n Ka ) Equation (60) can also be used in the case of the E-EMLM [25]. If we consider only the case of elastic service-classes [24]), then eq. (60) becomes: (60) φ (n,..., n,..., n K )= Cφ (n,..., n, n, n +,..., n K ) n i b i φ i (n,..., n i, n i, n i+,..., n K ), i K if n, K (6) Although the Marov chain becomes reversible by using φ (n) s, the existence of summations in eqs. (60) and (6) reveals that no PFS exists for the calculation of steady-state distribution P(n).

36 36 Ioannis Moscholios et al. C Determination of the average number of service-class calls To calculate the values of y (j ), we have, by definition that: P(n) y (j ) = n P(n nb = j ) = n G(j) n Ω j n Ω j (62) or y (j )G(j) = n Ω j n P (n) (63) In eq. (24) we have shown that: [ ] n i P (n) = α C i b i P (n l i ) B (i) n i m + C α j i b i P (n l i ) B m (i) for i K e l=0 m=l+ i K a l=0 m=l+ C < j T. Similarly, we can show for j C that: [ ] P (n) = n i n i α i b i P (n l i ) B m (i) + α i b i P (n l i ) B m (i) (64) j i K e l=0 m=l+ i K a l=0 m=l+ Therefore, we can write eq. (63) as follows: [ n i y (j)g(j)= n α i b i P (n l min(c, j) n Ω j i K e l=0 Initially, we consider the sum for i, when n i x: n Ω j n α i b i Since Ω j {ni x} = n i l=0 { P (n l i ) m=l+ hand side (RHS) of eq. (66) can be written as: i ) B m (i) + n i α i b i P (n l i ) B m (i) j m=l+ i K a l=0 m=l+ (65) B m (i) = n α i b i n Ω j n i x= P(n x i ) m=x n : s i n sb s + (n i x)b i = j xb i, n i x, n s 0, s i B (i) m (66) } the right ] n i n α i b i P(n x i ) B (i) n i m = n α i b i P (n x i ) n Ω j x= m=x x= n Ω j {n i x} m=x n i n α i b i P (n) B (i) n i P (n ) m =α i b i G(j xb i ) n G(j xb i ) x= n Ω j xbi m=x x= n Ω j xbi m=x j /b i α i b i G(j xb i )y (j xb i ) B m (i) x= m=x B (i) m = B (i) m = (67) Now we consider the sum for i=, when n x: n Ω j n α b n l=0 P(n l ) m=l+ B m () = n α b n Ω j n x= P(n x ) m=x B () m (68)

37 Batched Poisson Loss Model Supporting Elastic and Adaptive traffic 37 Since Ω j {n x}= (68) becomes: { n : s n sb s+(n x)b =j xb, n x, n s 0, s } the RHS of eq. n n α b P(n x ) B (i) n m = n α b P (n x ) B m (i) = n Ω j x= m=x x= n Ω j {n x} m=x n (n +x)α b P (n) B () n P (n ) m =α b G(j xb ) (n +x) B () G(j xb ) m x= n Ω j xb m=x x= n Ω j xb m=x j /b = α b G(j xb ) (y (j xb ) + x) B m () x= m=x (69) Equation (65), based on eq. (67) and eq. (69), can be written as eq. (46) (for an elastic service-class) or eq. (47) (for an adaptive service-class). References. Greenberg A, Sriant R (997) Computational Techniques for Accurate Performance Evaluation of Multirate, Multihop Communication Networs. IEEE/ACM Trans. Networing 5(2): Moscholios I, Logothetis M, Koinais G (2002) Connection dependent threshold model: A generalization of the Erlang multiple rate loss model. Performance Evaluation 48( 4): Shengye F, Wu Y, Suili F, Hui S (2004) Coordination-based optimisation of path bandwidth allocation for large-scale telecommunication networs. Computer Communications 27(): Moscholios I, Logothetis M, Koinais G (2005) Call-burst blocing of ON-OFF traffic sources with retrials under the complete sharing policy. Performance Evaluation 59(4): Vassilais V, Moscholios I, Logothetis M (2008) Call-level performance modelling of elastic and adaptive service-classes with finite population. IEICE Transactions on Communications E9-B(): Huang Q, Ko K-T, Iversen V (2008) Approximation of loss calculation for hierarchical networs with multiservice overflows. IEEE Transactions on Communications 56(3)