Evaluation of Multi-rate Loss Models for Elastic Traffic Vassilios G. Vassilakis, Ioannis D. Moscholios and Michael D. Logothetis Corresponding Author: Michael D. Logothetis WCL, Dept. of Electrical & Computer Engineering, University of Patras, 265 00 Patras, Greece, Tel.: +30-2610-996433; Fax: +30-2610-991855 E-mail:
[email protected] Abstract In this paper we review two extensions of the Erlang Multirate Loss Model (EMLM), whereby we can assess the call-level QoS of telecom networks supporting elastic service-classes. The first extension is the Connection Dependent Threshold Model (CDTM) where calls arriving at a communication link can reduce their bandwidth while increase their service time requirement, according to a set of thresholds indicating the link occupied bandwidth; the in-service calls do not alter the assigned bandwidth. The second extension is the Extended Erlang Multirate Loss Model (E-EMLM), where an arriving call at a communication link requires certain bandwidth and, if not available, the call can reduce its bandwidth requirement while increase its service time requirement; this is done in accordance with the bandwidth reduction of the in-service calls. For the evaluation of the two models, we compare the call-level QoS index of Call Blocking Probability (CBP), when the average allocated bandwidth of a link is the same for the two models. Our results show that if both stream and elastic traffic are present, the CDTM performs better than the E-EMLM for the elastic traffic. If only elastic services are accommodated in the link, the models perform equally well. Keywords: Quality of Service; Loss Model; Call Blocking; Elastic Traffic. 1. Introduction When service-classes of different characteristics are accommodated in a transmission link, one of the main concerns is the bandwidth allocation decision in order to achieve the best system performance and satisfy the QoS requirements of each service-class. We consider two types of traffic. The first is called stream traffic and requires some service guarantees (e.g real-time audio or video streaming). The second type is called elastic traffic and refers to applications that are able to adjust their rates according to the available link bandwidth (e.g file transfer). For the analysis of stream traffic in a single link loss system, the well-known Erlang Multirate Loss Model (EMLM) is used [1], [2]. In this model, Poisson arriving calls of service-classes with different bandwidth and service time requirements compete for the available link bandwidth under the complete sharing (CS) policy. The CS policy means that a call is accepted in the link if and only if the required bandwidth is available; otherwise the call is blocked and lost. An accepted call remains in the system for a service time, arbitrarily distributed. Later appeared models aspire to take into account the elastic services as well. In the single-retry model (SRM) [3], a blocked call is allowed to retry once, requesting for less bandwidth. In the single-threshold model (STM) [4], a call upon arrival requests for bandwidth according to the total link occupied bandwidth. Extensions of these models are the multi-retry (MRM) and multi-threshold (MTM) models, where calls can have multiple retrials and multiple thresholds, respectively [3], [4]. The Connection-Dependent Threshold Model (CDTM) [5] generalizes all the previously mentioned models by individualizing the thresholds among the service-classes. In the CDTM, the bandwidth of in-service calls does not alter, but, since calls can be accepted in the system with several bandwidth contingencies, the CDTM becomes appropriate for elastic services as well [6]. The Extended EMLM (E-EMLM) [7],[8] allows the acceptance of some calls that would be blocked in the EMLM because of the unavailability of the required bandwidth, by adjusting accordingly (reducing) the allocated bandwidth of the in-service calls. Besides, when a call departs from the system (having been serviced) then the in-service calls adjust again their bandwidth (the bandwidth of the in-service calls with reduced bandwidth is increased). Although a call requires a certain bandwidth initially, it can reduce it in order to be accepted in the system, while the in-service calls can alter their bandwidth; therefore, the EEMLM also becomes appropriate for elastic services. In this paper, first we review the CDTM and E-EMLM, emphasizing on the calculation of CBP which is recursive. The CDTM has not a product form solution and therefore the CBP determination is done approximately. The CBP calculation in the E-EMLM is accurate, since this model has a product form solution. Second, we comparatively evaluate them based on their CBP results in a single link. Since the
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CDTM is less accurate than the E-EMLM, in addition to the analytical CBP results we present simulation CBP results. Our evaluation shows that the CDTM performs better than the E-EMLM for elastic traffic, in respect of CBP, whereas the CDTM differentiates between stream and elastic traffic, holding the bandwidth of stream traffic always constant (this differentiation is not possible by the E-EMLM which assumes all service-classes as elastic services). This paper is organized as follows. Since the basis of both CDTM and E-EMLM is the EMLM, for presentation purposes, in Section 2, we describe the EMLM. In Section 3 and 4, we review the CDTM and E-EMLM, respectively, providing the recursive formulas for the CBP determination. Section 5 is the evaluation section. We present six application examples; in five of them two elastic services are accommodated in a link, whereas in the last example both elastic and stream services coexist in a link, for both models. We conclude in Section 6, discussing the extension of this work. 2. The Erlang Multirate Loss Model (EMLM) 2.1 Model description Assume a link of capacity C, where the C bandwidth units (b.u.) are commonly shared among Poisson arriving calls of K different service-classes; the arrival rate is denoted by λk (k = 1, …, K). System state (“micro-state”) is defined by the vector: n = (n1 ,n2 ,...,nk ,...,nK ) , where nk is the number of service-class k calls in the link. The system state space is denoted by Ω (n ∈ Ω). A call’s bandwidth requirement bk is satisfied if and only if at least bk b.u. of the link capacity are available upon arrival. Then the call is accepted and bk b.u. are seized by the call for a random (exponential) service time of mean µk-1 . Otherwise the call is blocked. 2.2 CBP Calculation The system occupancy j is defined as the total number of seized b.u. (“macro-state” of the system). The following recurrent formula is used for the calculation of the system occupancy distribution q(j): K
∑ αk bk q(j - bk )= jq(j),
j = 0,...,C
with q(x)=0 for x < 0 and
∑ q(j) = 1 ,
(1)
k=1
C
j=0
where αk =
λk µk-1 is
the offered traffic-load of service-class k.
The CBP of service-class k is given by: Bk =
C
∑
q(j) or Bk =
j=C-bk +1
bk -1
∑ q(C - j)
(2)
j=0
3. The Connection-Dependent Threshold Model (CDTM) 3.1 Model description Each arriving call of a service-class k may have Sk +1 bandwidth and service time requirements; one -1 initial requirement with values (bk ,µk-1 ) and Sk more requirements with values (bkcl ,µkc ) , l=1,…, Sk , where l
bkcS …>
-1 > µk-1 . µkc 1
The
-1 pair (bkcl ,µkc ) is l
used for service-class k when the link
occupied bandwidth at the call arrival is J kl-1 < j ≤ J kl , where J kl-1 and J kl are two successive thresholds of the service-class k, J kS +1 = C, while the highest possible (other than C) threshold is J kS = C- bkcS . By k k k -1 while the pair (bk ,µk-1 ) is used when j ≤ J k0 . A call cannot change the convention, bk = bkc0 and µk-1 = µkc 0 -1 assigned bandwidth while in service. The “total offered traffic-load” is equal for every pair (bkcl ,µkc ) and is l
defined as: “the product of the offered traffic-load by the required bandwidth per call” [9].
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Transmission Link
4 Bandwidth Requirements
b1c3
C J1 3
b1c2 b1c1
Service-classes s1, s2 3 thresholds for s1 2 thresholds for s2
Link Capacity
b1=b1c0
J1 J1 1 0
C
T h r e s h o l d s
J22 J21
3 Bandwidth Requirements
b2c2 b2c1 b2=b2c0
0
Fig. 1. Principles of the CDTM.
3.2 CBP Calculation The system occupancy distribution, q(j), is calculated by using the following approximate but recursive
formula: S
q(j)=
1 K 1 K k α k b k δk (j)q(j - bk ) + ∑ ∑ αkcl bkcl δkcl (j)q(j - bkcl ), ∑ j k=1 j k=1 l=1
with q(x)=0 for x < 0 and
j = 0 ,...,C
(3)
C
∑ q(j) = 1 , j=0
-1 where: αkcl = λk µkc and l
⎧⎪1 δk (j)= ⎨ ⎪⎩0
(when 1 ≤ j ≤ C and bkcl = 0) or (when j ≤ J k0 +bk and bkcl > 0) , otherwise
⎧⎪1 when ( J kl + bkcl ≥ j > J kl -1 + bkcl ) and (bkcl > 0) . δkcl (j)= ⎨ ⎪⎩0 otherwise
The CBP of service-class k is given by:
Bk =
C
∑
q(j)
(4)
j=C-bkcS +1 k
4. The Extended Erlang Multirate Loss Model (E-EMLM)) 4.1 Model description This is an extension of the EMLM aiming at allowing the acceptance of some calls that would be blocked in the EMLM. The main idea here is that when there is not enough available bandwidth, a newly arrived call, instead of being blocked, is accepted with reduced bandwidth. At the same time the bandwidth allocated to all in-service calls is also reduced. The bandwidth reduction of the newly arrived call and all inservice calls is the same and depends on the system state. The new bandwidth allocated to each call of service-class k in the state n is bk Φk ( n ) , where the factor Φk ( n ) expresses the state dependency (it is defined below). Because of the new values of the allocated bandwidth, the service time of each call is adjusted so that the product (service time) by (bandwidth per call) remains constant. The system occupancy j (“macro-state”) is defined as the sum of the initially required bandwidth of all in-service calls in the link. Thus j may have a value greater than the link capacity C, but up to a maximum value T, ( 0 ≤ j ≤ T ), (C ≤ T). The sum of the actual bandwidths (reduced or not) of all in-service calls is denoted by s (0 ≤ s ≤ C ) and is called “system allocation” [8].
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K
K
k =1
k =1
Thus at any time j = ∑ nk bk and s = ∑ nk bk Φk (n) . Both j and s in the E-EMLM coincide with the system occupancy j as it is defined in the EMLM, since for Φk ( n ) =1 the E-EMLM corresponds to the EMLM. The factor Φk ( n ) is defined as follows: When 0 ≤ j ≤ C, then Φk ( n ) = 1 (i.e. Φk ( n ) takes its maximum value; so the bandwidth allocated to each call of service-class k is the one requested initially ( bk ); in this case s = j ).
x( nk- ) (i.e. Φk ( n ) C), where nk− = (n1 , n2 ,..., nk −1 , nk − 1, nk +1 ,..., nK ) and the state multiplier (or weight) x(n) is given by the following recursion: When C < j ≤ T, then Φk ( n ) =
⎧ ⎪ ⎪ x( n )= ⎨ ⎪ ⎪ ⎩
1
for 0 ≤ j ≤ C
1 K nk bk x( nk- ) for C < j ≤ T ∑ C k=1 0
(5)
otherwise
If C < j ≤ T , then x(n), n ∈ Ω, satisfies the work conserving constraint C =
K
∑ nk bk
k =1
x( nk- ) [7], [8]. x( n )
In Fig.2 we show by a simple example what happens in the E-EMLM when a new call arrives to a transmission link of capacity C =5 b.u. and of maximum system occupancy T = 7 b.u. At a given time, only 2 b.u. of the link capacity are available, while the other 3 b.u. are seized by two other calls with bandwidth requirements b2 = 2 b.u. and b3 = 1.b.u. At this moment s = j = 3, so the bandwidth allocated to both calls is the initially required. At that time, assume that a new call requests b1 = 3 b.u. Although the new call requests more b.u. than are available in the link, it is accepted because b1 + j = 6 < T. After the call-acceptance, the bandwidth allocated to both in-service calls is reduced (together with the bandwidth allocated to the newly arrived call) and the system allocated bandwidth takes its maximum value s = 5 = C. At the same time the system occupied bandwidth becomes j = 6, thus a new call requesting b3 = 1.b.u. can be accepted, while the calls requesting more than 1 b.u. will be blocked. s = system allocation j = system occupancy C=5
C=5
T=7 Accepted Call b1Φ1( n )
T=7
s=5 j=6
s=3 j=3 Arriving Call b1 = 3
Fig. 2. Principles of the E-EMLM.
4.2
CBP Calculation The following recurrent formula calculates the system occupancy distribution q(j):
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K
∑ αk bk q(j - bk )= min(C, j )q(j),
j = 0 ,...,T
(6)
k=1
With q(x)=0 for x < 0 and
C
∑ q(j) = 1 j=0
The CBP of service-class k is given by:
Bk =
bk -1
∑ q(T - j)
(7)
j=0
5. Evaluation In this section we compare the two models in respect of the resultant CBP. The comparison/evaluation is performed as follows. We firstly get the analytical and simulation CBP results for the CDTM and then we determine the average link occupied bandwidth E(j). In the CDTM, the E(j) coincides with the average link allocated bandwidth E{s} (as it is defined in the E-EMLM). Afterwards, we define the maximum system occupancy T of the E-EMLM, so that the average link allocated bandwidth E{s} of the E-EMLM equals approximately to the average link occupied bandwidth E(j) of the CDTM; the simulation results are used for the E(j) of the CDTM. In order to facilitate the comparison/evaluation of the two models, for each model we calculate the average CBP, B, of all service-classes accommodated in the link, according to [9]: K
K
k=1
k=1
B = ∑ αk bk Bk / ∑ αk bk
(8)
Below, we present six application/numerical examples. For both the CDTM and E-EMLM we provide analytical CBP results, based on eq. (4) and eq. (7), respectively. Especially for the CDTM we also provide simulation CBP results, since the CDTM is an approximate model. Our simulation CBP results have been calculated as mean values from six runs. For each mean value a confidence interval of 95% has been defined. However, the resultant reliability ranges of our measurements are small enough and therefore in our graphs only the mean CBP values (without error bars) are presented. For each example we show in a figure the analytical results of both models together with the simulation CBP results of the CDTM versus the offered traffic-load. Besides we present in Tables (I to VI) the average link allocated bandwidth and average CBP of the service-classes versus the offered traffic-load for each model. In the same tables we also present the maximum system occupancy (T) for the E-EMLM . As a first example, we consider a transmission link of C = 30 b.u. that accommodates two elastic serviceclasses s1 and s2 , which require 3 and 7 b.u. per call initially, respectively. The offered traffic-load is α1 = 2 erl and α 2 = 0.5 erl for s1 and s2 , respectively. In the CDTM both service-classes are able to reduce their bandwidth unit by unit from 3 to 1 b.u. and from 7 to 1 b.u., respectively. More precisely, the thresholds of s1 are two: J 10 = 27 and J11 = 28, whereas of s2 are six: J 20 = 23, J 21 = 24, J 22 = 25, J 23 = 26, J 24 = 27 and J 25 = 28. That is, when the link occupied bandwidth, j, is less than or equal to 27 b.u. a call of serviceclass s1 is accepted in the link with its initial bandwidth requirement of b1 = b1c0 = 3 b.u. If 27 < j ≤ 28 (i.e. j = 28) then a call of service-class s1 is accepted in the link with b1c1 = 2 b.u. while if 28 < j ≤ 29 (i.e. j = 29) a call of service-class s1 is accepted in the link with b1c2 = 1 b.u. Similarly, a call of service-class s2 is accepted in the link with its initial bandwidth requirement of b2 = b2c0 = 7 b.u. when j ≤ 23, while it is accepted in the link with its last contingency bandwidth requirement of b2c6 = 1 b.u. when 28 < j ≤ 29 (i.e. j = 29). In the EEMLM, when j ≤ C, calls of service-classes s1 and s2 get their initially required bandwidth b1 = 3 and b2 =7 b.u. respectively. When C < j ≤ T, calls of s1 and s2 seize b1Φ1( n ) and b2Φ2 ( n ) b.u. respectively. We increase the traffic-load of each service-class and calculate the CBP based on eq. (4) and (7) for the CDTM and E-EMLM, respectively. We consider seven traffic-load points in the x-axis of Fig. 3, (1, 2,…, 7), by increasing simultaneously, the α1 and α 2 by 0.5 erl and 0.25 erl, respectively, so that the heaviest trafficload (7) is α1 = 5.0 erl and α 2 = 2.0 erl for s1 and s2 , respectively. This example is used in [8]. The CBP
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results of the CDTM are equal for both service-classes because their minimum required bandwidths are the same: b1c2 = b2c6 = 1 b.u. The results are shown in Fig. 3 and Table I. The second example differs from the previous only in service-class s1 , which reduces the required bandwidth from 3 to 2, based on one threshold: J10 = 28. The results are shown in Fig. 4 and Table II. The third example differs from the previous one only in the definition of the threshold of service-class s1 . The new threshold is: J10 = 24. The results are shown in Fig. 5 and Table III. As a fourth example, we consider a transmission link of C = 40 b.u. that accommodates two elastic service-classes s1 and s2 , which require 4 and 8 b.u. per call initially, respectively. The offered traffic-load is α1 = 1 erl and α 2 = 0.5 erl for s1 and s2 , respectively. In the CDTM both service-classes are able to reduce their bandwidth unit by unit from 4 to 2 b.u. and from 8 to 1 b.u., respectively. The thresholds of s1 are two: J10 = 35 and J11 = 37, whereas of s2 are seven: J 20 = 32, J 21 = 33, J 22 = 34, J 23 = 35, J 24 = 36, J 25 = 37 and J 26 = 38. We consider six traffic-load points in the x-axis of Fig. 6, (1, 2,…, 6), by simultaneously increasing the α1 and α 2 by 1.0 erl and 0.5 erl, respectively, so that the heaviest traffic-load (6) is α1 = 6.0 erl and α 2 = 3.0 erl for s1 and s2 , respectively. The results are shown in Fig. 6 and Table IV. The fifth example differs from the previous only in the definition of the thresholds of service-class s1 . The new thresholds are: J10 = 33 and J 11 = 35. The results are shown in Fig. 7 and Table V. As a sixth example, we consider a transmission link C=300 b.u. that accommodates 4 service-classes. In the CDTM the first two service-classes s1 and s2 correspond to stream traffic and require 1 and 6 b.u. per call, respectively. The other two service-classes s3 and s4 are elastic services, which require initially 6 and 24 b.u., but can reduce their bandwidth to 2 b.u. unit by unit, and to 8 b.u. in steps of 4, respectively. The thresholds of s3 are: J10 = 52, J11 = 56, J12 = 60 and J13 = 64. The thresholds of s4 are: J 20 = 32, J 21 = 36, J 22 = 40 and J 23 = 44. In the E-EMLM s1 , s2 , s3 and s4 , require initially 1, 6, 6 and 24 b.u. respectively. When the system occupancy j exceeds the link capacity C, the bandwidth of all in-service calls is reduced by a factor Φk ( n ) (k =1,…,4). We consider the following traffic-loads offered to the link: 60 erl for s1 , 12 erl for s2 and s3 , 1 erl for s4 . We increase the traffic-load of s1 from 60 to 120 erl in steps of 10, keeping constant the traffic-load of the other service-classes. This example comes from [5]. In Fig. 9 and 10 we show the analytical CBP results for both models and the simulation CBP results for the CDTM, for stream and elastic traffic, respectively.
Fig. 3. CBP results of the models versus the offered traffic-load for the first application example.
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Fig. 4. CBP results of the models versus the offered traffic-load for the second application example.
Fig. 5. CBP results of the models versus the offered traffic-load for the third application example.
Fig. 6. CBP results of the models versus the offered traffic-load for the fourth application example
Fig. 7. CBP results of the models versus the offered traffic-load for the fifth application example.
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Fig. 8. CBP results of stream traffic for the sixth application example.
Fig. 9. CBP results of the elastic traffic for the sixth application example.
Table I: First Example Average allocated bandwidth E{s} Traffic-load Max system Average CBP of both service-classes (%) (α1 , α2 ) (erl) CDTM(sim) CDTM(anal.) E-EMLM occupancy CDTM (sim) CDTM (anal.) E-EMLM (2.0, 0.5) 9.45 9.45 9.45 34 0.52 0,57 0.54 (2.5, 0.75) 12.51 12.49 12.52 35 1.91 2,07 1.82 (3.0, 1.0) 15.29 15.24 15.30 35 4.44 4,77 4.35 (3.5, 1.25) 17.71 17.62 17.73 35 8.03 8,48 7.91 (4.0, 1.5) 19.73 19.61 19.76 35 12.31 12,86 12.17 (4.5, 1.75) 21.39 21.24 21.44 35 16.95 17,52 16.76 (5.0, 2.0) 22.71 22.56 22.80 35 21.66 22,21 21.39
Table II: Second Example Average allocated bandwidth E{s} Traffic-load Max system Average CBP of both service-classes (%) (α1 , α2 ) (erl) CDTM(sim) CDTM(anal.) E-EMLM occupancy CDTM (sim) CDTM (anal.) E-EMLM (2.0, 0.5) 9.44 9.43 9.44 33 0.66 0.66 0.66 (2.5, 0.75) 12.48 12.43 12.45 33 2.26 2.28 2.35 (3.0, 1.0) 15.21 15.11 15.23 34 4.98 5.04 4.82 (3.5, 1.25) 17.59 17.41 17.59 34 8.63 8.70 8.64 (4.0, 1.5) 19.58 19.32 19.55 34 12.94 12.86 13.13 (4.5, 1.75) 21.22 20.87 21.15 34 17.61 17.18 17.88 (5.0, 2.0) 22.53 22.14 22.80 34 22.27 21.45 22.61
Table III: Third Example Average allocated bandwidth E{s} Traffic-load Max system Average CBP of both service-classes (%) (α1 , α2 ) (erl) CDTM(sim) CDTM(anal.) E-EMLM occupancy CDTM (sim) CDTM (anal.) E-EMLM (2.0, 0.5) 9.45 9.44 9.46 35 0.49 0.61 0.46 (2.5, 0.75) 12.53 12.47 12.52 36 1.77 2.17 1.43 (3.0, 1.0) 15.36 15.21 15.30 35 4.02 4.93 4.35 (3.5, 1.25) 17.85 17.58 17.73 35 7.24 8.70 7.91 (4.0, 1.5) 19.96 19.55 20.04 36 11.24 13.10 10.94 (4.5, 1.75) 21.70 21.17 21.77 36 15.70 17.78 15.45 (5.0, 2.0) 23.10 22.49 23.17 36 20.30 22.46 20.11
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Table IV: Fourth Example Average allocated bandwidth E{s} Traffic-load Max system Average CBP of both service-classes (%) (α1 , α2 ) (erl) CDTM(sim) CDTM(anal.) E-EMLM occupancy CDTM (sim) CDTM (anal.) E-EMLM (1.0, 0.5) 8.00 7.99 8.00 47 0.06 0.07 0.05 (2.0, 1.0) 15.80 15.74 15.78 47 1.36 1.61 1.36 (3.0, 1.5) 22.62 22.36 22.53 47 5.86 6.81 6.14 (4.0, 2.0) 27.80 27.25 27.55 47 13.18 14.83 13.90 (5.0, 2.5) 31.35 30.60 30.99 47 21.62 23.51 22.53 (6.0, 3.0) 33.68 32.85 33.29 47 29.89 31.57 30.65
Table V: Fifth Example Average allocated bandwidth E{s} Max system Average CBP of both service-classes (%) Traffic-load (α1 , α2 ) (erl) CDTM(sim) CDTM(anal.) E-EMLM occupancy CDTM (sim) CDTM (anal.) E-EMLM (1.0, 0.5) 8.00 8.00 8.00 47 0.05 0.06 0.05 (2.0, 1.0) 15.83 15.77 15.78 47 1.19 1.47 1.36 (3.0, 1.5) 22.76 22.47 22.97 48 5.22 6.36 4.30 (4.0, 2.0) 28.11 27.48 28.43 48 12.19 14.12 11.16 (5.0, 2.5) 31.78 30.92 32.16 48 20.59 22.71 19.59 (6.0, 3.0) 34.16 33.22 34.59 48 28.92 30.79 27.94
Table VI: Sixth Example Average allocated bandwidth E{s} Max system Average CBP of four service-classes (%) Traffic-load α1 (erl) CDTM(sim) CDTM(anal.) E-EMLM occupancy CDTM (sim) CDTM (anal.) E-EMLM 60 227.14 227.26 227.15 339 0.31 0.32 0.37 70 236.63 236.51 236.65 341 0.63 0.63 0.57 80 245.24 245.21 245.27 334 1.16 1.13 1.10 90 253.25 253.20 253.20 329 1,92 1.86 1.86 100 260.27 260.32 260.29 325 2.89 2.87 2.88 110 266.38 266.50 266.32 321 4.11 4.14 4.20 120 271.62 271.74 271.52 318 5.58 5.64 5.72
In the first example, we notice that for the lowest traffic-load (1), the CDTM has lower average CBP of both service-classes, B, than the E-EMLM, while for the other six traffic-loads (2-7) the E-EMLM performs better. In the second example, for the CDTM, the CBP of s1 is increased, while the CBP of s2 is decreased in comparison with the first example; however, the resultant B is increased. On the other hand, for the EEMLM, both service-classes increase their CBP in comparison with the first example. In the second example the CDTM gives better B than the E-EMLM for five traffic-loads (2 and 4-7). For one traffic-load (3), the EEMLM has better result. While for the lowest traffic-load (1), both models result in the same B. In the third example, by choosing a lower threshold for s1 , the resultant B of the two models changes significantly. For the CDTM, the CBP of s1 is reduced, while the CBP of s2 is increased, whereas a lower B results for the CDTM in comparison with the second example. For the E-EMLM, both service-classes reduce their CBP. This can be explained by the fact that, since the average allocated bandwidth E{s} (which is chosen to be the same for both models) increases, the maximum system occupancy (T) also experiences an increment. Consequently, the resultant B for the E-EMLM is lower than it was in the previous example. Comparing B between the two models (Table III), the CDTM gives better results for two traffic-loads (3 and 4), while the E-EMLM performs better for five traffic-loads (1, 2 and 5-7). Similar results are obtained from the fourth and fifth examples, where the total offered traffic-load of both service-classes is the same. Hence from all these examples, we observe that the two models behave equally well. By choosing different thresholds or alternating the number of bandwidth contingencies for some service-classes in the CDTM, while selecting the values of the maximum system occupancy in the EEMLM accordingly, we can obtain better results either by the CDTM or by the E-EMLM. In the sixth example, the CBP of stream traffic ( s1 and s2 , Fig.8) is higher in the CDTM than in the EEMLM, on the contrary to the elastic traffic ( s3 and s4 , Fig.9), where the CDTM gives lower (better) CBPs, for all traffic-loads. This is due to the fact, that the E-EMLM does not differentiate between stream and elastic traffic. Thus, calls belonging to s1 and s2 , violate the stream nature of the service-classes and reduce their bandwidth by a factor Φ1( n ) and Φ2 ( n ) respectively, when needed. Comparing B, both models perform equally well.
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6.
Conclusions We review the CDTM and E-EMLM, and evaluate them by comparing the resultant CBP either of stream or elastic traffic, as well as the average CBP of all service-classes accommodated in a single link. In presence of both stream and elastic traffic, the CDTM performs better than the E-EMLM for elastic traffic, while, the E-EMLM performs better than the CDTM for the stream traffic, violating, however, the nature of the stream traffic by allocating a reduced bandwidth to stream-type calls. In presence only of elastic traffic, the models perform equally well, as far as the average CBP of all service-classes is concerned. Extended Work: For a complete comparison of the two models, some other system parameters could be examined such as the average number of calls in the link, the average throughput and the average service time per service-class. An important extension is to evaluate the two models under QoS guarantee requirements. A further extension is to combine the CDTM and E-EMLM and produce a new model that would describe well the behavior of elastic services in the presence or not of stream traffic. In the new model, calls of different service-classes will arrive at a transmission link either with several bandwidth contingencies or fixed bandwidth requirements, while the in-service calls will be able to adjust their bandwidth in order for a new call to be accepted in the link.
Acknowledgment Work supported by the research program Caratheodory of the Research Committee of the University of Patras.
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