The Erlang multirate loss model with Batched Poisson arrival processes under the bandwidth reservation policy

Computer Communications 33 (2010) S167–S179

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The Erlang multirate loss model with Batched Poisson arrival processes under the bandwidth reservation policy Ioannis D. Moscholios a, Michael D. Logothetis b,* a b

Department of Telecommunications Science and Technology, University of Peloponnese, Tripolis 22100, Greece WCL, Department of Electrical and Computer Engineering, University of Patras, 265 00 Patras, Greece

a r t i c l e

i n f o

Article history: Available online 5 May 2010 Keywords: Batched Poisson process Bandwidth reservation QoS Time-call congestion Markov chain

a b s t r a c t The Erlang Multirate Loss Model (EMLM) has been widely used as a springboard in the study of multirate loss systems. In this paper, we study an extension of the EMLM in order to ensure QoS guarantee per service-class in the heterogeneous environment of telecom networks. The proposed model is named Batched Poisson EMLM under Bandwidth Reservation (BR) policy (BP-EMLM/BR), since its input process is a Batched Poisson arrival process; that is, calls of each service-class arrive to a fixed-capacity link as batches. The distribution of the batch size can be general. The batch blocking discipline is the partial batch blocking, i.e. depending on the available link bandwidth, a part of an arriving batch can be accepted while the rest of it is discarded. The Call Admission Control is based on the BR policy, whereby we can achieve our goal to guarantee specific QoS for each service-class at call-level. In order to apply the BR policy in the BPEMLM/BR, we examine two methods also used in the EMLM: (a) the Roberts’ method and (b) the Stasiak– Glabowski (S&G) method. The proposed model does not have a product form solution and therefore we propose approximate but recursive formulas for the calculation of various performance measures, such as time and call congestion probabilities as well as link utilization. A comparison between the analytical and simulation results shows that both methods provide satisfactory results when equalization between time congestion probabilities of all service-classes is desired. The S&G method, however, performs much better than the Roberts’ method in the case where more than two service-classes are accommodated in the link. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The Erlang Multirate Loss Model (EMLM) has been widely used as a springboard in the study of multirate loss systems, both of wired and wireless networks [1–7], due to the following features:  The steady-state probabilities have a Product Form Solution (PFS) which leads to an accurate calculation of the main QoS performance index, the Call Blocking Probabilities (CBP), [8,9].  The recursive calculation of macro-state probabilities (link occupancy distribution); a feature that facilitates the CBP calculation and broadens the EMLM’s applicability range to systems of large capacities [9–11]. A quite interesting extension of the EMLM, among many others ([12–17]), is the incorporation of the batch Poisson call-arrival process. In this process simultaneous call-arrivals (batches) do occur at time-points which follow a negative exponential distribution [18–21]. In these papers the importance of the batch Poisson process is highlighted due to the fact that: (a) calls arrive as batches in * Corresponding author. Tel.: +30 2610 996433; fax: +30 2610 991855. E-mail address: [email protected] (M.D. Logothetis). 0140-3664/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2010.04.039

many applications and (b) ‘‘peaked” and ‘‘bursty” arrival processes can be readily represented, in an approximate way, with the aid of the peakedness factor, z. This factor is the ratio of the variance over the mean of the number of arrivals; if z = 1, the arrival process is Poisson; if z < 1, the arrival process is quasi-random [22]; if z > 1, the process is more peaked and bursty than Poisson (e.g. overflow traffic [19]). The EMLM with Batch Poisson arrivals (BP-EMLM) is examined in [18] under the hypothesis of partial batch blocking and geometric batch size distribution, while a general batch size distribution is considered in [19]. According to the partial batch blocking discipline a part of the entire batch (one or more calls) is accepted in the system while the rest of it is discarded, when the available link bandwidth is not enough to accommodate the entire batch. In the case of the geometric batch size distribution, the BP-EMLM coincides with the Delbrouck’s model [23]. The latter generalizes the EMLM since it allows the call-arrival process to have different peakedness factors. In both [18] and [19] the model has a PFS, and the link occupancy distribution is determined via an accurate recursive formula. An asymptotic analysis of [19] is presented in [20]. Furthermore, the models of [18,19] are extended in [21] to include: (a) a fixed routing network (sequence of links), (b) batches that arrive in a state-dependent manner, (c) the complete batch

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blocking discipline, where the whole batch is blocked and lost, even if only one call of the batch cannot be accepted in the system, and (d) the upper limit and guaranteed minimum bandwidth sharing policies. In this paper we propose an extension of the BP-EMLM of [19] (partial batch blocking – generally distributed batch size) in order to ensure QoS guarantee per service-class in the heterogeneous environment of telecom networks. The proposed model is named Batched Poisson EMLM under Bandwidth Reservation (BR) policy (BP-EMLM/BR). According to the BR policy, a fraction of the available link bandwidth (denoted as BR parameter) is reserved to benefit services with higher bandwidth per call requirements, because they receive worse call-level QoS than other services with less bandwidth per call requirements. Besides, the BP-EMLM/BR can be used in alternate route systems to model overflow traffic. In this case, the BR policy (also known as trunk reservation) protects direct traffic in high-usage routes from overflow traffic [22]. The BR policy destroys the PFS of the BP-EMLM and, therefore, the calculation of various performance measures, such as Time Congestion (TC) and Call Congestion (CC) probabilities, or link utilization, cannot be accurate; for these calculations, approximate but still recursive formulas are proposed. In the BP-EMLM/BR, we examine the incorporation of the BR policy according to: (a) the Roberts’ method [24] and (b) the Stasiak–Glabowski (S&G) method [25], already proposed for the EMLM under the BR policy (EMLM/BR). A simple recursive approximation for the calculation of the link occupancy distribution is proposed in [24]. Its simplicity is based on the assumption that the population of calls of a service inside its Reservation Space (RS) is negligible; the RS of a service is the fraction of the system state-space where the access of new calls is denied. In general, the Roberts’ approximation is satisfactory, but it may become critical when CBP equalization among services is desired. In this case, Stasiak and Glabowski proposed a more accurate method (but more complex), where the population of calls of a service inside its RS is taken into account in the CBP calculation [25]. In the BPEMLM/BR, we show that both methods provide satisfactory results but the S&G method gives better results in some cases and especially when the offered traffic-load is high. Furthermore, we examine the selection of BR parameters to equalize TC or CC probabilities. This paper is organized as follows: In Section 2 we review the BP-EMLM. In Section 3, we propose the BP-EMLM/BR. In Sections 3.1 and 3.2 we present the Roberts’ method and the S&G method, respectively; in both sections, we propose formulas for the calculation of various performance measures. In Section 4 we present numerical results in order to evaluate both methods by comparing analytical with simulation results. We conclude in Section 5.

2. Review of the BP-EMLM

GðjÞ ¼

8 > < 1P

Pbj=bk c b k K k¼1 k bk l¼1 B l1 Gðj

1 >j

:

a

for j ¼ 0;  lbk Þ for j ¼ 1; . . . ; C;

0

ð1Þ

otherwise;

where:

ak = kk/lk, bj/bkc is the largest integer less than or equal to j/bk and b k is the complementary batch size distribution given by B l b k ¼ P1 Bk . B r¼lþ1 r l

If Bkr ¼ 1 for r = 1 and Bkr ¼ 0 for r > 1 we have a Poisson process and the EMLM results:

GðjÞ ¼

8 > < 1P > :

K k¼1 k bk Gðj

1 j

a

for j ¼ 0;  bk Þ for j ¼ 1; . . . ; C;

0

ð2Þ

otherwise:

Having recursively determined G(j)’s in the BP-EMLM various other performance measures can be calculated [19]:  The average number of service k calls given that the system state is j, denoted as E(nkjj), is determined by:

Eðnk jjÞ ¼ ak

bX j=bk c

, k b B l1 Gðj  lbk Þ GðjÞ:

ð3Þ

l¼1

 The occupied link bandwidth in state j is given by:

j¼

K X

bk Eðnk jjÞ:

ð4Þ

k¼1

 The average number of service k calls in the system, denoted as  k , is calculated via: n

k ¼ n

C X

Eðnk jjÞGðjÞ:

ð5Þ

j¼1

 CC probability of service k, denoted as Cbk, is the probability that a new service k call is blocked and is given by:

 . bk  n bk;  k ak B C bk ¼ ak B

ð6Þ

b k denotes the average size of service k arriving batches where: B b k ¼ P1 rBk . and is given by: B r r¼1  TC probability of service k, denoted as Pbk, is the probability that at least C  bk + 1 b.u are occupied and is given by: C X

P bk ¼

G1 GðjÞ;

ð7Þ

j¼Cbk þ1

P where: G ¼ Cj¼0 GðjÞ is a normalization constant.  The link utilization, denoted as U, can be determined via Eq. (1):

U¼

C X

jGðjÞ:

ð8Þ

j¼1

We consider K services accommodated in a link of fixedcapacity of C bandwidth units (b.u.). Calls of each service k (k = 1, . . . , K) require bk b.u., compete for the available link bandwidth under the Complete Sharing (CS) policy (all call types compete for all bandwidth resources) and have an exponentially distributed service time with mean l1 k . Calls arrive to the link according to a batch Poisson process with arrival rate kk and follow the partial batch blocking discipline. The batch size distribution Bkr is the probability that there are r calls in an arriving batch of service k. The steady-state probabilities in the BP-EMLM are given by a PFS which leads to an accurate recursive formula for the calculation of the link occupancy distribution, G(j), where j is the total occupied link bandwidth [19]:

If calls of service k arrive in batches of size sk where sk is given by the geometric distribution with parameter bk, i.e. Pr ðsk ¼ r Þ ¼ ð1  bk Þbr1 with r P 1, then the BP-EMLM coincides with the model k proposed by Delbrouck in [23], as shown in [18,19]. More precisely, b k ¼ bl , Eq. (1) takes the form: since B l

GðjÞ ¼

8 > < 1P 1 >j

:

0

K k¼1 k bk

a

Pbj=bk c l¼1

for j ¼ 0; bl1 Gðj  lbk Þ for j ¼ 1; . . . ; C;

ð9Þ

otherwise:

The geometric distribution is a memoryless distribution and a discrete equivalent of the exponential distribution [26]. Because of these features it is highly used as a batch size distribution. In the

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Fig. 1. Difference between the CC and TC probability (BP-EMLM).

b k ¼ 1 in Eq. case of the geometric batch size distribution, replace B 1bk (6). Prior to the application of the BR policy in the BP-EMLM it is important to note the distinction between TC and CC probabilities. These probabilities coincide in the case of Poisson arrivals (Poisson Arrivals See Time Averages, PASTA property [22]]), but not in the case of batch Poisson arrivals (unless Bkr ¼ 1 for r = 1 and Bkr ¼ 0 for r > 1 which is a Poisson process); when coincide they are commonly referred as CBP. TC probability is determined by the proportion of time the system is congested. An observer, who is not part of the system, can measure this probability. CC probability is determined by the proportion of arriving calls that find the system congested. An observer who is part of the system (i.e. an arriving call) can measure this probability. Herein, we present the difference between TC and CC probabilities. Consider a link of capacity C = 12 b.u. that accommodates calls of two services with requirements b1 = 1 and b2 = 2 b.u., respectively. In Fig. 1, we assume that the link has 6 b.u. available for new calls. At the first time-point, a 2nd service call arrives. This call is accepted and the available link bandwidth reduces to 4 b.u. At the second time-point, a 1st service batch arrives consisting of five calls. One out of these calls is blocked and lost. This type of blocking has to do with CC probability (one blocking event, Fig. 1 ‘‘CC probability calculation”). At the third time-point, we assume that there is available bandwidth to accept the whole 1st service batch (two calls) while at the fourth time-point we assume that the whole 2nd service batch (two calls) is discarded. This type of blocking has to do with both TC probability (one blocking event, Fig. 1 ‘‘TC probability calculation”) and CC probability (two blocking events, Fig. 1 ‘‘CC probability calculation”). At the fourth to sixth time-points, the comments are similar and thus they are omitted.

RS consists of the states j = C  tk + 1, . . . , C. This assumption facilitates an approximate calculation of G(j)’s. To illustrate Roberts’ assumption consider, for example, a link of C = 4 b.u which accommodates K = 2 services whose calls require b1 = 1 and b2 = 2 b.u., respectively. The BR parameters of both services are chosen t1 = 1, t2 = 0 so that b1 + t1 = b2 + t2 (for TC probabilities equalization). Fig. 2 presents the one-dimensional Markov chain for our example. The two different types of arrows correspond to the two different services and show all the possible transitions between the states j = 0, . . . , 4. For example, while in state j = 0 a 1st service batch may arrive consisting of one, two or three calls and therefore the system ‘‘will pass” to states j = 1, j = 2 or j = 3, respectively. Similarly, if the system is in state j = 0 and a 2nd service batch arrives consisting of one or two calls the system ‘‘will be transferred” to states j = 2 or j = 4, respectively. As one notices, the 1st service calls are not allowed to enter their RS, which consists of state j = 4 only. It is because of this prohibition that Roberts’ assumption is reasonable. To include in Eq. (1) the assumption that the population of service k calls with tk > 0 is negligible inside the RS of service k, we denote the variable:

 Dk ðj  bk Þ ¼

In the BP-EMLM/BR a new service k call is accepted in the system if and only if j + bk 6 C  tk where tk is the BR parameter of service k which denotes the available link b.u reserved from service k to benefit the other services. The application of the BR policy in the BP-EMLM destroys the PFS of the model as a simple example in [27] shows. In what follows we examine the applicability of the BR policy in the BP-EMLM according to the Roberts’ method and the S&G method.

when j 6 C  tk ; when j > C  t k :

ð10Þ

Therefore Eq. (1) takes the form:

GðjÞ ¼

8 > < 1P 1 >j

:

K k¼1 k Dk ðj

a

for j ¼ 0; Pbj=bk c b k  bk Þ l¼1 B l1 Gðj  lbk Þ for j ¼ 1; . . . ; C;

0

otherwise: ð11Þ

Bkr

3. The proposed BP-EMLM/BR

bk 0

Bkr

Note: If ¼ 1 for r = 1 and ¼ 0 for r > 1 then the EMLM/BR results. The average number of service k calls given that the system state is j, E(nkjj), is given by:

( Eðnk jjÞ ¼

ak 0

. Pbj=bk c b k when j 6 C  t k ; l¼1 B l1 Gðj  lbk Þ GðjÞ when j > C  tk :

3.1. The Roberts’ method Roberts assumes that the population of service k calls, which require bk b.u. while tk > 0, is negligible inside the RS of service k. The

Fig. 2. One-dimensional Markov chain (Roberts’ method).

ð12Þ

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Based on Eqs. (11) and (12) we calculate the various performance measures according to Eqs. (4)–(6) and Eq. (8). To determine the TC probabilities we modify the bounds of Eq. (7) in order to include the BR parameter tk: C X

P bk ¼

G1 GðjÞ;

ð13Þ

j¼Cbk t k þ1

where: G ¼

PC

j¼0 GðjÞ

is a normalization constant. Fig. 4. Calls of three services of bandwidth b1, b2 and b3 respectively, with b1 > b2 > b3, transfer calls of a fourth service to a state j inside its RS.

3.2. The S&G method Stasiak and Glabowski show that calls of a service k with tk > 0 may exist inside their RS. In the previous example, assume now that there are under service two calls of the 1st service (system state j = 2). If a 2nd service batch arrives consisting of r calls then only one call is accepted in the system while the rest r  1 calls are discarded. In that case the system ‘‘is transferred” in state j = 4 (since b2 = 2 b.u.) where there can still be the two 1st service calls under service. This can be seen as a transfer of 1st service calls from state j = 2 to j = 4 due to a 2nd service call-arrival. Therefore it is possible to have 1st service transitions from state j = 4 to j = 3 as Fig. 3 shows. In a more complicated example there might be many services whose call-arrivals help in transferring calls of a certain service inside its RS (see Fig. 4). For the calculation of the service k calls population inside its RS, we denote by E*(nkjj) the average number of service k calls in state j which belongs to the RS of service k and follow the analysis of [25]. We denote by wk, i(j) a weight factor that determines the portion of E*(nkjj) which is transferred to state j (that belongs to the RS of service k) by a service i call, where i – k. Since more than one service i can contribute in the transfer of service k calls to the RS of service k, we use different weights to show that this contribution can be different for each service i (where i – k), [25]:

, wk;i ðjÞ ¼ ai bi

K X

a j bj :

ð14Þ

Fig. 5. The S&G method in the BP-EMLM under the BR policy.

j¼1;j – k

The calculation now of E*(nkjj) can be based on the following formula:

8 P . b k Gðj  lbk Þ GðjÞ when j 6 C  t k ; < ak bj=bk c B l1 l¼1  E ðnk jjÞ ¼ P : K  when j > C  tk : i¼1;i – k E ðnk jj  bi Þwk;i ðjÞ

GðjÞ ¼ ð15Þ

Based on Eq. (4) if we modify E(nkjj) then the corresponding value of j should also be modified. Therefore, based on Eqs. (15) and (4) we calculate the modified occupied link bandwidth, j*, for every state j: 

j ¼

K X

bk E ðnk jjÞ:

1  >j

:

K k¼1

for j ¼ 0; Pbj=bk c b k ak bk l¼1 B l1 Gðj  lbk Þ for j ¼ 1; . . . ; C;

0

ð17Þ

otherwise:

The entire procedure of the S&G method is shown in Fig. 5. The TC probabilities can be calculated by Eq. (13). Based on Eqs. (15)–(17)  k , the CC one can calculate the average number of service k calls, n probabilities and the link utilization according to Eqs. (5), (6) and (8), respectively.

ð16Þ

k¼1

Having calculated both E*(nkjj) and j* we determine the modified values of G(j)’s based on:

Fig. 3. One-dimensional Markov chain (S&G method).

8 > < 1P

4. Numerical results – evaluation We compare the analytical results of BP-EMLM/BR obtained by the Roberts’ and S&G methods in respect of: the TC probabilities, the CC probabilities and the link utilization. As standard we take simulation results (mean values of seven runs, 95% confidence interval). For comparison we include the analytical results obtained by the BP-EMLM. The first example is the same as the one presented in [21]. A link of capacity C = 60 b.u. accommodates two services whose calls require b1 = 1 b.u. and b2 = 12 b.u., respectively. We consider the following sets of BR parameters: (1) t1 = 3 and t2 = 0, (2) t1 = 7 and t2 = 0 and (3) t1 = 11 and t2 = 0. The third set equalizes the TC probabilities of both services since b1 + t1 = b2 + t2. The batch size of both services follows the geometric distribution with parameters: b1 = 0.2 and b2 = 0.5. The call holding time is exponentially distrib1 uted with mean l1 1 ¼ l2 ¼ 1. The values of the offered traffic are:

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a1 = 10 erl and a2 = 2 erl. In the 1st column of Tables 1–10, a1 remains constant while a2 decreases by 0.2. Table 1 shows, as a reference, the analytical results of TC and CC probabilities (for both services) and link utilization in the case of the BP-EMLM (CS policy). In Tables 2–4 we present, for both services and each set of BR parameters, the analytical and simulation results for the TC probabilities when the Roberts’ and S&G methods are applied. In Tables 5–7 we present, for both services and each set of BR parameters, the analytical and simulation results for the CC probabilities when the Roberts’ and S&G methods are applied. Table 8 presents, for each set of BR parameters, the analytical results of the link utilization for both methods, while Table 9 presents the corresponding simulation results. According to the results of Tables 1–9 we observe that:  The BR policy is not fair to 1st service calls, because the decrease of TC and CC probabilities of the 2nd service is less significant compared to the increase of TC and CC probabilities of the 1st service. This results in the decrease of the link utilization; compare the link utilization results of Table 1 (CS policy) with those of Table 9 (BR policy).  The S&G method gives slightly better results than the Roberts’ method, compared to simulation results. This is because only the 1st service has a positive BR parameter and therefore only

Table 4 Analytical and simulation results of the equalized TC probabilities for the BR policy (t1 = 11).

a2

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

Eq. TC probabilities (%) (Roberts’ method)

Eq. TC probabilities (%) (S&G method)

Simulation (%) 1st service

2nd service

21.40 19.12 16.78 14.37 11.94 9.51 7.13 4.88

21.45 19.16 16.81 14.40 11.96 9.52 7.14 4.89

22.32 ± 0.25 19.87 ± 0.22 17.39 ± 0.16 14.80 ± 0.18 12.35 ± 0.18 9.80 ± 0.12 7.35 ± 0.05 5.04 ± 0.14

22.30 ± 0.23 19.89 ± 0.14 17.42 ± 0.22 14.79 ± 0.29 12.38 ± 0.23 9.78 ± 0.19 7.38 ± 0.09 5.05 ± 0.13

the population of 1st service is assumed to be negligible (in the RS: C  t1 + 1, . . . , C). Such an assumption does not lead to a high approximation error. Because of the higher complexity of the S&G method together with our extensive study on examples with two services we suggest the usage of Roberts’ method when only two services are considered.  Although the third set of BR parameters achieves the TC probabilities equalization, the same set cannot achieve the CC probabilities equalization (see Table 7). According to Eq. (6), which

Table 1 Analytical results of TC and CC probabilities and link utilization for the CS policy. a2

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

TC probabilities (%) (analytical results)

CC probabilities (%) (analytical results)

1st service

2nd service

1st service

2nd service

2.52 2.20 1.88 1.57 1.26 0.97 0.70 0.46

25.56 22.99 20.32 17.54 14.69 11.79 8.92 6.16

3.14 2.74 2.35 1.96 1.58 1.22 0.88 0.58

47.84 45.17 42.26 39.08 35.62 31.85 27.76 23.38

Link utilization (analytical results)

37.14 35.84 34.38 32.72 30.84 28.70 26.26 23.46

Table 2 Analytical and simulation results of the TC probabilities for the BR policy (t1 = 3). a2

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

TC probabilities (%) (Roberts’ method)

TC probabilities (%) (S&G method)

Simulation (%)

1st service

2nd service

1st service

2nd service

1st service

2nd service

8.27 7.23 6.20 5.18 4.18 3.22 2.33 1.53

24.44 21.98 19.42 16.76 14.04 11.27 8.53 5.89

8.28 7.24 6.20 5.18 4.18 3.23 2.33 1.53

24.45 21.99 19.43 16.77 14.04 11.27 8.53 5.89

7.46 ± 0.08 6.50 ± 0.10 5.56 ± 0.13 4.75 ± 0.08 3.83 ± 0.09 2.95 ± 0.07 2.12 ± 0.07 1.40 ± 0.05

24.82 ± 0.19 22.32 ± 0.32 19.61 ± 0.19 16.80 ± 0.26 14.27 ± 0.23 11.35 ± 0.23 8.68 ± 0.23 6.26 ± 0.22

Table 3 Analytical and simulation results of the TC probabilities for the BR policy (t1 = 7).

a2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

TC probabilities (%) (Roberts’ method)

TC probabilities (%) (S&G method)

Simulation (%)

1st service

2nd service

1st service

2nd service

1st service

2nd service

13.88 12.24 10.58 8.93 7.30 5.71 4.19 2.81

23.24 20.87 18.40 15.86 13.25 10.62 8.03 5.53

13.91 12.27 10.61 8.95 7.31 5.71 4.20 2.81

23.27 20.90 18.43 15.87 13.27 10.63 8.03 5.54

14.90 ± 0.13 13.10 ± 0.23 11.16 ± 0.22 9.54 ± 0.14 7.65 ± 0.10 6.07 ± 0.08 4.35 ± 0.10 2.93 ± 0.04

24.05 ± 0.18 21.55 ± 0.18 19.16 ± 0.21 16.45 ± 0.20 13.94 ± 0.31 10.98 ± 0.19 8.46 ± 0.20 5.82 ± 0.26

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Table 5 Analytical and simulation results of the CC probabilities for the BR policy (t1 = 3).

a2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

CC probabilities (%) (Roberts’ method)

CC probabilities (%) (S&G method)

Simulation (%)

1st service

2nd service

1st service

2nd service

1st service

2nd service

8.74 7.65 6.56 4.43 5.49 3.42 2.48 1.63

47.06 44.45 41.61 35.13 38.51 31.44 27.46 23.17

8.75 7.66 6.57 4.44 5.49 3.43 2.48 1.63

47.07 44.45 41.62 35.13 38.51 31.45 27.46 23.17

8.23 ± 0.07 7.17 ± 0.10 6.13 ± 0.14 4.22 ± 0.12 5.23 ± 0.08 3.25 ± 0.07 2.33 ± 0.08 1.56 ± 0.03

47.06 ± 0.18 44.41 ± 0.14 41.77 ± 0.14 35.16 ± 0.10 38.39 ± 0.21 31.61 ± 0.25 27.48 ± 0.22 23.49 ± 0.25

Table 6 Analytical and simulation results of the CC probabilities for the BR policy (t1 = 7).

a2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

CC probabilities (%) (Roberts’ method)

CC probabilities (%) (S&G method)

Simulation (%)

1st service

2nd service

1st service

2nd service

1st service

2nd service

14.36 12.69 10.99 9.29 7.60 5.96 4.39 2.95

46.22 43.66 40.88 37.84 34.54 30.95 27.06 22.88

14.40 12.72 11.02 9.31 7.62 5.97 4.40 2.95

46.24 43.68 40.89 37.86 34.55 30.95 27.06 22.88

15.57 ± 0.14 13.68 ± 0.23 11.71 ± 0.22 10.02 ± 0.13 8.05 ± 0.09 6.41 ± 0.08 4.60 ± 0.10 3.11 ± 0.06

46.50 ± 0.05 43.80 ± 0.12 41.12 ± 0.13 37.93 ± 0.18 34.77 ± 0.06 31.10 ± 0.21 27.28 ± 0.22 23.18 ± 0.26

Table 7 Analytical and simulation results of the CC probabilities for the BR policy (t1 = 11).

a2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

CC probabilities (%) (Roberts’ method)

CC probabilities (%) (S&G method)

Simulation (%)

1st service

2nd service

1st service

2nd service

1st service

2nd service

22.16 19.82 17.41 14.93 12.42 9.91 7.44 5.11

44.93 42.42 39.69 36.75 33.55 30.08 26.35 22.35

22.20 19.86 17.44 14.96 12.44 9.92 7.45 5.11

44.96 42.45 39.72 36.76 33.56 30.09 26.35 22.35

22.97 ± 0.27 20.37 ± 0.22 17.86 ± 0.16 15.36 ± 0.18 12.58 ± 0.26 10.23 ± 0.11 7.58 ± 0.06 5.32 ± 0.08

45.13 ± 0.19 42.54 ± 0.14 39.89 ± 0.25 36.71 ± 0.28 33.54 ± 0.20 30.05 ± 0.20 26.36 ± 0.32 22.47 ± 0.26

Table 8 Analytical results of the link utilization for the BR policy.

a2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

t1 = 3

t1 = 7

t1 = 11

Roberts’ method

S&G method

Roberts’ method

S&G method

Roberts’ method

S&G method

36.82 35.54 34.10 32.47 30.63 28.52 26.12 23.36

36.81 35.54 34.10 32.47 30.63 28.52 26.12 23.36

36.52 35.25 33.83 32.22 30.40 28.33 25.96 23.24

36.50 35.24 33.82 32.22 30.40 28.33 25.96 23.24

36.16 34.90 33.48 31.89 30.09 28.04 25.71 23.04

36.14 34.88 33.47 31.88 30.08 28.04 25.71 23.04

Table 10 Offered traffic-load versus CC probabilities equalization and the corresponding BR parameters.

Table 9 Simulation results of the link utilization for the BR policy.

a2

a2

t1 = 3

t1 = 7

t1 = 11

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

36.82 ± 0.06 35.42 ± 0.05 34.06 ± 0.09 32.38 ± 0.06 30.62 ± 0.09 28.58 ± 0.06 26.12 ± 0.09 23.36 ± 0.08

36.17 ± 0.04 34.95 ± 0.13 33.58 ± 0.11 32.05 ± 0.08 30.20 ± 0.07 28.22 ± 0.09 25.87 ± 0.11 23.22 ± 0.07

35.77 ± 0.30 34.62 ± 0.14 33.33 ± 0.08 31.74 ± 0.09 29.97 ± 0.05 28.00 ± 0.08 25.66 ± 0.11 23.03 ± 0.10

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

CC probability (Roberts’ method) (%) 1st service

2nd service

42.22 38.68 37.05 32.87 30.38 27.07 24.22 19.88

42.33 39.87 36.86 34.05 30.66 27.37 23.81 20.15

BR parameter t1

22 22 23 23 24 25 27 29

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holds when the BR policy is applied to the BP-EMLM, the calculation of CC probabilities depends on the offered traffic-load rather than on the BR parameters. For a certain offered trafficload one can find the BR parameters which almost equalize CC probabilities. When the offered traffic-load changes, then a new set of BR parameters is needed. To show it, we present

in Table 10 the BR parameters that almost equalize the CC probabilities of the services (they are obtained by the Roberts’ method). The second example shows in a convincing way that the S&G method gives better results than the Roberts’ method when TC

0.20

sim.

0.19

BR 3rd set

0.18

st

Time congestion probabilities (1 service)

0.17

S&G

Roberts’

0.16 0.15 0.14 0.13

S&G

0.12

Roberts’

0.11 0.10

BR 2nd set

sim.

0.09 0.08

S&G

0.07 0.06

BR 1st set

Roberts’

0.05 0.04

sim.

0.03 0.02 0.01

CS 3.0

3.5

4.0

4.5 α1 (erl)

5.0

5.5

6.0

Fig. 6. TC probabilities of the 1st service.

0.20 0.19

BR 3rd set

sim.

0.18

S&G

nd

Time congestion probabilities (2 service)

0.17

Roberts’

0.16

Roberts’

0.15

BR 2nd set

S&G

0.14

sim.

0.13 0.12

S&G

0.11

BR 1st set

0.10 0.09

Roberts’

sim.

0.08

CS

0.07 0.06 0.05

3.0

3.5

4.0

4.5 α1 (erl)

5.0

Fig. 7. TC probabilities of the 2nd service.

5.5

6.0

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probabilities equalization is required. We consider a link of capacity C = 100 b.u. which accommodates three services whose calls require b1 = 1 b.u., b2 = 4 b.u. and b3 = 16 b.u., respectively. As far as the BR parameters are concerned we examine three different sets: (1) t1 = 5, t2 = 4 and t3 = 0, (2) t1 = 10, t2 = 8 and t3 = 0 and (3) t1 = 15, t2 = 12 and t3 = 0. The third set equalizes the TC probabilities of all services since b1 + t1 = b2 + t2 = b3 + t3. The batch size of all services follows the geometric distribution with parameters: b1 = 0.75,

b2 = 0.5 and b3 = 0.2. The call holding time is exponentially distrib1 1 uted with mean l1 1 ¼ l2 ¼ l3 ¼ 1. The values of the offered traffic are: a1 = 6, a2 = 4 and a3 = 2 erl. In the x-axis of all figures, a2 and a3 remain constant while a1 decreases by 0.5. Fig. 6 presents the analytical and simulation results of the TC probabilities of the 1st services for both policies; in the case of the BR policy we consider both methods and all sets of BR parameters. Figs. 7 and 8 present the corresponding results

0.32

CS

0.31 0.30

BR 1st set

0.29

S&G sim.

0.27

Roberts’

0.26

BR 2nd set

0.25 0.24

sim.

0.23

S&G

0.22

Roberts’

0.21 0.20

BR 3rd set

sim.

0.19 0.18

S&G

0.17

Roberts’

0.16 0.15 3.0

3.5

4.0

4.5 α1 (erl)

5.0

5.5

6.0

5.5

6.0

Fig. 8. TC probabilities of the 3rd service.

MRAE for the time congestion probabilities (%)

rd

Time congestion probabilities (3 service)

0.28

17.0 16.0 15.0 14.0 13.0 12.0 11.0 10.0 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0

S & G (1 st set o f B R p aram eters) R o b erts' (1 st set o f B R p aram eters) S & G (2 n d set o f B R p aram eters) R o b erts' (2 n d set o f B R p aram eters)

3.0

3.5

4.0

4.5 α1 (erl)

5.0

Fig. 9. MRAE for the TC probabilities.

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st

Call congestion probabilities (1 service)

for the 2nd and the 3rd service, respectively. According to Figs. 6–8 the S&G method gives always better results when the 3rd set of BR parameters is used (TC probabilities equalization). In the case of the 1st and the 2nd sets the S&G method gives worse results in

0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05

Figs. 6 and 7 and better results in Fig. 8. To conclude which method performs better when the 1st and the 2nd sets are used, we present in Fig. 9 the Mean Relative Approximation Error (MRAE) of TC probabilities, given by:

BR 3rd set

sim.

S&G S&G

Roberts’ sim.

BR 2nd set Roberts’

S&G BR 1st set sim.

Roberts’ CS

3.0

3.5

4.0

4.5 α1 (erl)

5.0

5.5

6.0

Fig. 10. CC probabilities of the 1st service.

0.30 0.29 0.28

BR 3rd set

sim.

0.27

S&G

nd

Call congestion probabilities (2 service)

0.26 0.25

Roberts’

sim.

BR 2nd set

0.24 0.23 0.22 0.21

Roberts’

0.20

S&G

BR 1st set

S&G

0.19 0.18

sim.

0.17 0.16

CS

0.15

Roberts’

0.14 0.13 0.12 0.11

3.0

3.5

4.0

4.5 α1 (erl)

5.0

Fig. 11. CC probabilities of the 2nd service.

5.5

6.0

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PK MRAE ¼

k¼1 RAEk

K

100%;

ð18Þ

where RAEk stands for the Relative Approximation Error for each service k given by:

 .   RAEk ¼ Pbk;an  Pbk;sim  Pbk;sim ;

ð19Þ

where Pbk, an, Pbk, sim are the TC probabilities obtained by Eq. (13) and by simulation, respectively.

0.40

CS

0.39 0.38 0.37

BR 1st set S&G

0.35

sim.

0.34

BR 2nd set

Roberts’

0.33

S&G

0.32

sim.

0.31 0.30

Roberts’

0.29

BR 3rd set

0.28 0.27 0.26

Roberts’

0.25

sim.

0.24

S&G

0.23 3.0

3.5

4.0

4.5 α1 (erl)

5.0

5.5

6.0

Fig. 12. CC probabilities of the 3rd service.

MRAE for the call congestion probabilities (%)

rd

Call congestion probabilities (3 service)

0.36

3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9

S&G (1 st set of BR parameters) Roberts' (1 stnd set of BR parameters) Roberts'nd(2 set of BR parameters) S&G (2 set of BR parameters)

3.0

3.5

4.0

4.5 α 1 (erl)

5.0

Fig. 13. MRAE for the CC probabilities.

5.5

6.0

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Sum of MRAE (%)

According to Fig. 9 the S&G method is slightly worse than the Roberts’ method and therefore the S&G method is worth applied only when TC probabilities equalization is required. Fig. 10 presents the analytical and simulation results of the CC probabilities of the 1st service for both policies; in the case of the BR policy we consider both methods and all sets of BR parameters. Figs. 11 and 12 present the corresponding results for the 2nd and the 3rd service, respectively. According to Figs. 10–12 the S&G method gives always better results when the 3rd set of BR parameters is used. In the case of the 1st and the 2nd sets it is uncertain which method performs better and therefore we present in Fig. 13 the MRAE of CC probabilities. According to Fig. 13 the S&G method is slightly worse than the Roberts’ method when the 1st set of BR parameters is used, and slightly better in the case of the 2nd set of BR parameters. In Fig. 14 we present the sum of the MRAE of the TC

19.0 18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.0

probabilities (Fig. 9) and the CC probabilities (Fig. 13). The results (for the 1st and 2nd set of BR parameters) show that the S&G method performs slightly worse. In Fig. 15 we give the simulation results of the link utilization for the CS and BR policy (all three different sets). The results show that an increase in BR parameters decreases the link utilization. The worst link utilization appears when TC equalization is required, since the decrease of TC and CC probabilities of the 3rd service is less significant compared to the increase of TC and CC probabilities of the 1st and 2nd service. Finally, in Fig. 16, we present the analytical and simulation results of the TC probabilities for the BR policy and the 3rd set of BR parameters, when the link capacity increases from 100 b.u. to 200 b.u. (in steps of 20 b.u.). As reference values, we present the corresponding analytical results of the CS policy. According to Fig. 16, the S&G method performs better than the Roberts’ method

S&G (1st set of BR parameters) Roberts' (1st set of BR parameters) S&G (2nd set of BR parameters) Roberts' (2nd set of BR parameters)

9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 3.0

3.5

4.0

4.5 α1 (erl)

5.0

5.5

6.0

Fig. 14. MRAE of TC probabilities + MRAE of CC probabilities.

74 CS 1st set of BR parameters 2nd set of BR parameters 3rd set of BR parameters

73 72

Link utilization (b.u.)

71 70 69 68 67 66 65 64 3.0

3.5

4.0

4.5 α1 (erl)

5.0

Fig. 15. Simulation results of the link utilization.

5.5

6.0

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0.33 0.32 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22

Time congestion probabilities

0.21

CS (3rd service)

BR 3rd set

0.20 0.19 0.18 0.17 0.16

S&G

0.15 0.14 0.13 Roberts’

0.12 0.11 0.10 0.09 0.08

CS (2nd service)

0.07 0.06 0.05 0.04 0.03

CS (1st service)

0.02 0.01 0.00 100

120

140

160

180

200

Capacity (b.u) Fig. 16. TC probabilities versus link capacity.

when C is between 100 and 140 b.u (e.g. for C = 140, the S&G method gives 6.26%, the Roberts’ method 6.15%, while the simulation result is 6.30% ± 0.17%). As C increases farther, the TC probabilities decrease and the two methods give almost the same results (e.g. for C = 200, the S&G method gives 0.517%, the Roberts’ method 0.513% while the simulation result is 0.520% ± 0.04%). For such small values of the TC (or CC) probabilities it is better to apply the Roberts’ method whose complexity is much lower. 5. Conclusion We propose the application of the BR policy in the BP-EMLM with the partial batch blocking discipline and a generally distrib-

uted batch size. The importance of BP-EMLM/BR is not only that we can guarantee specific call-level QoS for each service, but also that we can apply the BP-EMLM/BR to model overflow traffic in alternate route systems, where the BR policy is a must in order for the direct traffic in high-usage routes to be protected from overflow traffic. To apply the BR policy we investigate the Roberts’ and the S&G methods, already proposed in the EMLM/BR. Since the BP-EMLM/BR does not have a PFS, we propose approximate but recursive formulas for the calculation of the TC and CC probabilities or the link utilization. A comparison between the analytical and simulation results shows that: (a) both methods provide satisfactory results in all circumstances, (b) the Roberts’ method is preferable when only two service-classes exist, or when the offered

I.D. Moscholios, M.D. Logothetis / Computer Communications 33 (2010) S167–S179

traffic-load is low and (c) the S&G method could be considered when CBP equalization is required, whereas it is preferable when more than two services exist and the offered traffic-load is high. References [1] M. Logothetis, S. Shioda, Medium-term centralized virtual path bandwidth control based on traffic measurements, IEEE Trans. Commun. 43 (10) (1995) 2630–2640. [2] A. Greenberg, R. Srikant, Computational techniques for accurate performance evaluation of multirate, multihop communication networks, IEEE/ACM Trans. Networking 5 (2) (1997) 266–277. [3] M. Logothetis, G. Kokkinakis, Path bandwidth management for large scale telecom networks, IEICE Trans. Commun. E83-B (9) (2000) 2087–2099. [4] H. Shengye, Y. Wu, F. Suili, S. Hui, Coordination-based optimisation of path bandwidth allocation for large-scale telecommunication networks, Comput. Commun. 27 (1) (2004) 70–80. [5] P. Fazekas, S. Imre, M. Telek, Modelling and analysis of broadband cellular networks with multimedia connections, Telecommun. Syst. 19 (2002) 263– 288. [6] D. Staehle, A. Mäder, An analytic approximation of the uplink capacity in a UMTS network with heterogeneous traffic, in: Proceedings of the 18th International Teletraffic Congress (ITC), Berlin, 31st August–5th September 2003, pp. 81–90. [7] A. Mäder, D. Staehle, Analytic modeling of the WCDMA downlink capacity in multi-service environments, in: ITC Specialist Seminar on Performance Evaluation of Wireless and Mobile Systems, August 31–September 02, 2004, pp. 217–226. [8] C. Chigan, R. Nagarajan, Z. Dziong, T. Robertazzi, On the capacitated loss network with heterogeneous traffic and contiguous resource allocation constraints, in: Proceedings of the Advanced Simulation Technologies Conference, Seattle, Washington, 2001. [9] J.S. Kaufman, Blocking in a shared resource environment, IEEE Trans. Commun. 29 (10) (1981) 1474–1481. [10] J.W. Roberts, A service system with heterogeneous user requirements, in: G. Pujolle (Ed.), Performance of Data Communications Systems and their Applications, North Holland, Amsterdam, 1981, pp. 423–431. [11] Z. Dziong, J.W. Roberts, Congestion probabilities in a circuit switched integrated services network, Perform. Eval. 7 (4) (1987) 267–284.

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