Performance Analysis of ATM Routing with Nonlinear Equivalent Capacity: Symmetric Case
C. Aswakul and J. Barria
[email protected], [email protected] Department of Electrical and Electronic Engineering Imperial College of Science, Technology and Medicine
University of London, London SW7 2BT, U.K.
Abstract— This paper presents a performance analysis of ATM rout- ing with nonlinear equivalent capacity in symmetric and fully connected network. The conventional dynamic alternative routing (DAR) is here ex- tended to the environment of multiple services with nonlinear equivalent capacity. This extension uses the trunk reservation policy and dynamic ser- vice/route separation. The analytical model of generalized DAR is formu- lated by a reduced-load approximation and then solved by a linear approxi- mation model and traffic grouping. The analytical results are here found to be in good agreement with 95% confidence interval of discrete-event sim- ulations. To solve the analytical model, the required time complexity is
and the space complexity is , where is the link capacity and is the number of capacity-guaranteed services. Best-effort services are also considered in the comparison of the generalized DAR when the connection admission control (CAC) is based on either nonlinear equivalent capacity or standard peak-rate assignment. The reported numerical results suggest that the CAC with nonlinear equivalent capacity can increase the total mean revenue rate of the generalized DAR by up to for the scenarios of single capacity-guaranteed service and for the scenarios of three capacity-guaranteed services.
I. INTRODUCTION
With the connection-oriented feature of asynchronous trans- fer mode (ATM) networks, well-known multirate (or multiser- vice) loss network studies have been widely recognized as an important framework to tackle ATM routing problems (e.g., [1], [2], [3]). Most of the reported works in loss networks assume the linear relationship between the number of connections and the amount of link capacity required by the con- nections. In this paper, function " will be referred to as the equivalent capacity1.
The need to deal with nonlinear equivalent capacity in a loss network framework arose only recently, with the novel admis- sion/scheduling schemes under service separation, route separa- tion and multiplexing across routes in [3]. Under those schemes, the equivalent capacity " for each service (and possibly for each route) is monotone increasing and concave to reflect the statistical multiplexing effect of cell streams sharing the same buffer. The impact of this nonlinearity on ATM routing is twofold. Firstly, routing algorithms that work well in the con-
The work of C. Aswakul was supported by Anandamahidol Foundation, Thai- land. C. Aswakul is currently at Electrical Engineering Department, Faculty of Engineering, Chulalongkorn University, Thailand ([email protected]).
#
This terminology -equivalent capacity- was first introduced in [4] under the situation of a first-in-first-out ATM multiplexer accessed by 2-state fluid sources. However, this terminology is used in a wider sense in this paper to generally define the amount of link capacity that is required for a given number of connections to satisfy all the cell-level quality of service requirements. This terminology is also known as thecapacity functionin [3].
ventional networks with linear equivalent capacity may not per- form equally well in the nonlinear domain. Secondly, since most of the existing numerical techniques are solely based on linear equivalent capacity, it becomes non-trivial how to ana- lyze the routing models that must be formulated in the nonlinear domain.
To balance between spreading and packing routes in the non- linear domain, this paper identifiesdynamic alternative routing (DAR) [5] as a promising candidate. The nature of DAR is to fill an alternative route between each node pair with as many connections as possible before a blocking occurs on that alter- native route, and hence a packing feature. In response to the blocking, a new alternative route is randomly chosen (for future uses) from the whole set of alternative routes between the node pair, and hence a learning mechanism, which allows a spread- ing feature. In other words, DAR has the feature of spreading calls in batches, each of which enables the economies of scale from the nonlinear equivalent capacity. Furthermore, DAR en- joys the advantages of implementation simplicity since DAR is decentralized and requires only local information to make rout- ing decisions.
However, to the best of our knowledge, DAR has only been proposed for the telephone network, in which a single (tele- phone) service with linear equivalent capacity needs to be con- sidered [5]. The aim of this paper is thusto extend the conven- tional DAR into the scenario of multiple services with nonlinear equivalent capacity. With respect to this aim, it is desirable to have both an analytical model and numerical techniques to solve the model. Furthermore, since the whole point of this extension is due solely to the nonlinear equivalent capacity, a question arises on what will happen if one simply ignores this nonlin- earity, e.g., when the standard peak-rate assignment is used for connection admission control (CAC). It is then interesting to compare the performance of DAR as obtained from CAC with the nonlinear equivalent capacity and CAC with the peak-rate assignment. While the equivalent-capacity CAC may be capa- ble of accepting more connections for the capacity-guaranteed services, the peak-rate CAC should be able to accept more con- nections for the best-effort services. Therefore, both service types need to be considered, should one desire a reasonable comparison framework.
In order to address the extension of DAR and its benefits over the DAR with peak-rate assignment, this paper simplifies the
problem by exploiting the network symmetry. The symmet- ric features are here comprised of (i) a fully connected net- work structure, (ii) an identical traffic load for every origin- destination node pair and (iii) an identical capacity for every link. Extension to asymmetric network scenarios will be re- ported in a forthcoming paper.
II. EXTENSION OFDYNAMICALTERNATIVEROUTING
A. Routing Operation
Fig. 1 depicts a diagram showing the routing operation of DAR with multiple services as naturally extended from the con- ventional DAR. Within a fully connected network, each pair of origin node and destination node has a current tandem node assigned toeachservice . A fresh call of service arriving at the node pair is routed on the direct route (via the direct link between the node pair) whenever possible. If the fresh call is blocked from its direct route, then a two-link alter- native route will be tried via the current tandem node associ- ated with the call’s service type. If the call is also blocked from any of the links on the alternative route, then the call is lost and a new tandem node is randomly selected for this service from the set of all nodes excluding the call’s origin and destination.
The current tandem node remains unchanged if the call is admit- ted. Note that call routing of a given service does not depend on the current tandem nodes of other services and no alternative routes with more than two links are allowed. In practice, the fully connected network structure can be easily built up by allo- cating capacities to ATM virtual paths—which can be viewed as logical links for the virtual channel routing (e.g., [6]). Having the fully connected structure also decreases the (virtual channel) call setup time and improves the network reliability.
Direct RouteR1
Origin Node
Destination Node
Current Tandem Node for Overflow Calls of Services
Ts
O D
Alternative RouteR2
Fig. 1. Diagram of DAR operation when service- calls arrive at a node pair.
B. Connection Admission Control Operation
To define CAC operation, suppose that each link, with the capacity , is used by (capacity-guaranteed) services and routes. For and , let denote the number of ongoing connections of service on route and define as the amount of link capacity required by the
connections. Based on the dynamic service separation with
dynamic route separation2[3] and the trunk reservation policy [7], CAC operation is defined as follows.
CAC Policy Definition: Calls of service on route are blocked from accessing a given link on the route if and only if
" $
% & (
% & ) (1)
where is the trunk reservation parameter assigned by that link for service- calls using route .
The trunk reservation parameters can be assigned in various ways. For instance, to allow directly routed calls a full-capacity access on the link, can be set to / "
for a direct route . Further, to put some restriction to the link access of alternatively routed calls, for an alterna- tive route can be set to a constant greater than . Note that can depend on the number of ongoing connections
if " is nonlinear (although we write here rather
than for notational convenience).
C. Analytical Model
This subsection presents an analytical model to capture the dynamics of capacity-guaranteed services for the generalized DAR. Let 3 denote the number of nodes and let 4 5 de- note mean of the exponentially distributed holding times of (capacity-guaranteed) service- calls. By network symmetry, each link has the same set of loading, trunk reservation and hence the probability of call blocking. Let7 8 8 and9 8 de- note the arrival rate, trunk reservation parameter and blocking probability of the service- calls being offered to each directly routed link. Since the considered network is fully connected, each link receives overflow calls via a total of: 3 " : alter- native routes. For service- calls arriving at the considered link on each alternative route, let7 < < and9 < denote their ar- rival rate, trunk reservation parameter and blocking probability.
The trunk reservation parameter from (1) is thus denoted here either by 8 for a direct route or by < for an alterna- tive route , in order to distinguish the role of trunk reservation on direct and alternative routes.
Let us now state two assumptions essential for the analysis of network routing by the reduced-load approximations (e.g., [1], [3], [8]).
Assumption 1:Calls of each service arrive at a link on each route according to an independent Poisson process.
Assumption 2: Calls of each service on a given route are blocked independently on each link of that route.
It follows immediately from Assumptions 1 and 2 that
7 < 7 8 9
8
" 9 <
3 " :
(2)
@
This paper presents the dynamic service separation with dynamic route sep- aration because this scheme results in the most computationally intensive case to analyse. Other schemes like static separation or multiplexing across routes require less computational effort and can be easily analysed by the techniques developed in this paper.
for . The arrival rate7 < thus depends on the block- ing probabilities9 8 and9 < . Conversely, by Assumption 1, the blocking probabilities9 8 and9 < can be calculated from7 < and other known parameters via the model of single-link system (to be further elaborated in Section III). Here, let us refer to the calculation of this single-link system as procedure and write it in a functional form as
(3)
where 9 8 9
$
8 9
<
9 $< 7 8 7 $8
7 < 7 $<
8
$8
<
$<
5 5 $
and " $ "
Combining (2) and (3) finally results in a set of fixed-point equations, which can be solved by a direct repeated substitution as given in Recursion 1. Here, represents the approximate value of at theth iteration for : and de- notes the initial value for the recursion.
Recursion 1 1. Set .
2. Calculate 7 < for from (2) with being re- placed by .
3. Solve for a new estimate from (3) (see Recursion 2).
4. Update / from step 3.
5. If / " 4 / ) (tolerance parame- ter), then stop the recursion. Otherwise, set / and go back to step 2.
After the convergence of Recursion 1 (empirically within 20 iterations), the overall blocking probability of service- calls, denoted by9 !8, between a pair of origin and destination nodes can be calculated from
9
! 8 9 8 " " " 9
<
$ (4)
The most intensive computation required by Recursion 1 is typically at step 3. This step involves the procedure of single- link system. Consequently, the complexity of procedure greatly affects the overall complexity of recursion, especially the computational time. The following section focuses on pro- cedure and how to reduce its computational complexity.
III. SINGLE-LINKSYSTEM ANDTRAFFICGROUPING
Consider now any single link in the symmetric network. Let
%
denote the total number of traffics accessing this link. By Assumption 1, all these traffics, indexed by & % in this section, are independent Poisson processes. Since the call holding times of each traffic are assumed to be independent and exponentially distributed, the resultant single-link system is ex- actly the system being analyzed by the linear approximation modelin [9] under the complete sharing policy. The principle of linear approximation model is to convert the problems from the nonlinear domain of equivalent capacity into the linear domain so that the efficient numerical techniques in the linear domain can be employed.
In order to extend the linear approximation model to imple- ment the procedure with trunk reservation policy, three mod- ifications are here needed. Firstly, the trunk reservation param- eter (whose value can depend on the link state) must be dealt with. Secondly, the Roberts’s recursive algorithm [7] is used in- stead of the Kaufman-Roberts’s recursive algorithm [10], [11]
(which is used in [9]) to solve for the blocking probabilities.
The Roberts’s recursive algorithm is employed because of the empirically observed good accuracy that can be obtained from this algorithm [7] or its extended versions [12], [13]. Having an accurate algorithm in the linear domain should enable us to focus on the intrinsic error incurred by the linear approxima- tion model itself. Thirdly, a traffic grouping concept will be additionally introduced to decrease the involved computational complexity.
Recursion 2 summarizes how procedure is implemented.
Here, for traffic& & % , let( ) ) " , ) and9 ) de- note the number of connections, the equivalent capacity, the offered load, and the blocking probability. Further, let 9 ) represent the approximate value of 9 ) at the th iteration for
: and let 9 ) denote the initial value for the recursion. For the parameters of linear approximation model, define0 ), 1 and )3< 4 6 7 as the capacity required by a call of traffic&, the reduced link capacity and the approximate trunk reservation parameter of traffic&, respectively.
Recursion 2 1. Set .
2. Calculate: ( ) for all& from: ( ) , ) " 9 ) . 3. Calculate, ;=) >: ( )@ and, A=) >: ( )@ / for all&. 4. Calculate the parameters of linear approximation model (0 ) 1 and )3<4 6 7, for all&) from0 ) ) C, A=)E " ) C, ;=
)E
1 " H J
)& C, A=) ) C, ;=)E " , ;=) ) C,
A=
)E E and )3< 4 6 7
) : L where ) : L is the value of trunk reser- vation parameter of traffic & at the mean link state : L
: ( : ( J
.
5. Solve for the blocking probabilities9 ) for all& in the linear approximation model by the Roberts’s recursive algorithm [7].
6. Update9 ) / 9 ) from step 5, for all&.
7. If 9 ) / " 9 ) 4 9 ) / ) M (tolerance parame- ter) for all&, then stop the recursion. Otherwise, set / and go back to step 2.
For the results in Section V, Recursion 2 is found to converge within approximately 20 iterations. Recursion 2 requires the time complexity of % and the space complexity of % . Since
% : 3 " and Recursion 2 is the most intensive
computational part in Recursion 1, it follows that Recursion 1 requires the time complexity of 3 and the space com- plexity of 3 .
To reduce the computational complexity, the concept of traf- fic grouping is now proposed. The idea is to group together the traffics of thesameservice on different routes. In general, grouping traffics in a single-link system with nonlinear equiv- alent capacity might not be as trivial as initially thought. This is because the nonlinearity destroys the additive property, i.e.,
Link Traffics in
Subset Traffics in Subset
A A
'
Switch
(a) Original System:X
Traffics in Link SubsetA
'
Switch
(b) System After Traffic Grouping:Y Aggregate Traffic a
Fig. 2. Diagram depicting the concept of traffic grouping.
/
/ . For this reason, traffic grouping has to be done in the linear approximation model, i.e., at step 5 in Recursion 2.
Formally, consider a single link with the finite capacity accessed by independent Poisson traffics. Partition the set of these traffics into two subsets and . Without loss of generality, all traffics in will be grouped together to form a new aggregate traffic . Figs. 2(a) and (b) depict the original system and its version after this traffic group- ing. For convenience, name these systems and , respec- tively. Consider now the traffic parameters in the linear ap- proximation model. Let 7 3
7
) 5 3 7
) 0 3 7
) 3 7
) 9 3 7
) and
7 3 7
) 5 3 7
) 0 3 7
) 3 7
) 9 3 7
) denote, for a traffic& in sys- tems and the arrival rate, the reciprocal of mean holding time, the capacity requirement by each call, the trunk reserva- tion parameter, and the blocking probability, respectively.
Regarding traffic grouping, if the following conditions hold:
& 7 3 7
< H ) 7 3 7
) && 5 3 7
< 5 3 7
) 0 3 7
< 0 3 7
)
3 7
< 3 7
) for all & , and &&& 7 3
7
) 7 3 7
)
5 3 7
) 5 3 7
) 0 3 7
) 0 3 7
) 3 7
) 3 7
) for all& ; then we have that9 3
7
< 9 3 7
) for all& and9 3
7
) 9 3 7
)
for all& . In other words, under conditions & –&&& , all traffics in can be grouped together to form a new aggregate traffic (traffic ) without affecting the computed blocking prob- abilities. The explanation is immediately obtainable from the superposition theorem and memoryless property of independent Poisson processes.
After establishing the conditions for traffic grouping, focus now on how to reduce the magnitude of% . Specifically, in the linear approximation model for the symmetric network, all the
: 3 " : alternatively routed traffics of each service can be
grouped together to form an aggregate new traffic. The arrival rate of the aggregate traffic is: 3 " : 7 < for service- calls, where . The new value of% after traffic grouping becomes : accounting for directly routed traffics and an-
other aggregate alternatively routed traffics. Using this traffic grouping concept, Recursion 1 requires the time complexity of
and the space complexity of , significantly re- duced and independent of the number of nodes3 .
IV. REVENUECOMPARISONFRAMEWORK
In this section, we compare the performance of generalized DAR when CAC with either the nonlinear equivalent capacity or the standard peak-rate assignment is in use. The system perfor- mance is here measured in terms of the obtainable mean revenue rate (the long-run average rate at which the revenues are ob- tained with the unit in, say, dollars per second). The mean rev- enue rates are here derived first for capacity-guaranteed services and then for best-effort services. Since the capacity-guaranteed services have been indexed by in Section II, let us index best-effort services by
For , let denote the revenue generated in one time unit by an ongoing connection of capacity-guaranteed ser- vice . The unit of for is, e.g., dollars per second per (ongoing) connection. The mean revenue rate of capacity-guaranteed service from all3 3 " pairs of origin and destination nodes in the network can then be directly calculated from
7 8 3 3 " " 9 !8
5 (5)
where9 !8 is obtainable from (4).
Let ! denote the average capacity that is utilized by a connection of capacity-guaranteed service , i.e., !
$%& ' ( ) / "
. For each link in the symmetric network, let+ denote the average capacity that is utilized by all capacity-guaranteed services. Then, by taking into account all the routes on each link, it follows immediately that
+ $
%& ! 7 8 " 9 8 / : 7 < 3 " : " 9
<
5 (6)
where9 8 9 < 7 < can be obtained from Recursion 1.
Recall now that best-effort services can use only the capacity that is not utilized by any capacity-guaranteed services. Each link has this unutilized capacity of the amount " + on average.
For simplicity, suppose that the considered symmetric network transforms each unit of this unutilized capacity into the revenue of best-effort services with a constant rate , . The unit of , is, e.g., dollars per second per capacity unit. With all the unutilized capacity on3 3 " links, the mean revenue rate of best-effort services from this symmetric network can then be written as
,3 3 " " + (7)
In this paper, we propose that the mean revenue rate from the generalized DAR with either the peak-rate CAC or the equivalent-capacity CAC should be compared in terms of
bounds at the minimum value of , as well as at the maximum value of , . The minimum value of , is 0, i.e., when only the capacity-guaranteed services are considered. To find the maximum value of , , we use the fact that best-effort services, due to their inferior quality of service, should not be expected to give higher mean revenue rate than any capacity-guaranteed services, given the same amount of available link capacity. One connection of a capacity-guaranteed service on a link requires the capacity of at most due to the concavity of " . This connection produces the mean revenue rate of . If the capacity is used by best-effort services instead, then the obtainable mean revenue rate will be , , which must not exceed . It then follows that , 4 Considering all capacity-guaranteed services , we have
, & %
$
(8)
Finally, the total mean revenue rate, denoted by , from both capacity-guaranteed services and best-effort services in the network can be obtained from H
$& ,
V. NUMERICALRESULTS
Numerical results in this section are aimed at evaluating two issues, namely, (i) the validity of analytical model as formulated for the generalized DAR and (ii) the comparative study of the formulated DAR with CAC based on either the nonlinear equiv- alent capacity or the standard peak-rate assignment. For all the reported results,3 and 4 5 (time unit) for all . Like [9], the equivalent capacity / is adopted for capacity-guaranteed service and some examples of are plotted in Fig. 3. The peak rate for each connection of service is set to . The average capacity utilized by each connec- tion of service is set to !
$%& ' ( ) / "
. The revenue rate is set to for capacity-guaranteed service . From (8), the range of , is then be- tween and& % $ 4 . For all the results reported here, the total mean revenue rate is obtained from both , and , . The specific settings here do not restrict the applicability of the analytical model and numerical techniques proposed in this paper.
Both analytical and simulation results are obtained. The ana- lytical results are calculated from the derivations in Sections II to IV. For the discrete-event simulation, both Assumptions 1 and 2, used in deriving for the analytical results, are not made (except for the application of Assumption 1 to fresh call ar- rivals). To measure 95% confidence intervals, the standard method of batch means is used. The simulation settings have been chosen such that the resultant width of confidence inter- vals is comparative to the obtained analytical results.
A. Scenarios of Single Capacity-Guaranteed Service
This subsection considers two scenarios each with a different capacity-guaranteed service . The settings and results
0 10 20 30 40 50 60 70 80 90 100
0 50 100 150 200 250
Number Of Connectionsn EquivalentCapacity)(Mbits/s)Gns(
( , ) = ( 1 , 14.1421 ) Mbits/sa b ( , ) = ( 2 , 7.0711 ) Mbits/sa b ( , ) = ( 3 , 0 ) Mbits/sa b
Fig. 3. Equivalent capacity of the form .
0 10 20 30 40 50 60 70 80 90 100
0 0.5 1 1.5 2
x 1010
Load Offered To Each Node Pair (Erlang)
TotalMeanRevenueRate:(Units/s)W
DAR With Peak-Rate CAC (Analytical) DAR With Equivalent-Capacity CAC (Analytical) Simulation ( 95% Confidence Interval )
r0= 1
r0= 1
r0= 0 r0= 0
Fig. 4. Peaky source scenario ( # " $ , ,
( ) #* # # , ( ) #. / ,
all in Mbits/s;2
#
* is varied;
peakedness " $ ).
can be found in Fig. 4 for thepeaky source scenarioand in Fig 5 for thesmooth source scenario3.
Figs. 4 and 5 suggest a very good agreement between the ob- tained analytical results and the corresponding 95% confidence intervals as measured from the simulation. There are two impli- cations. Firstly, Assumptions 1 and 2 seem to be justified in this case. Secondly, the linear approximation model—employed to solve for all the analytical results of DAR with the equivalent- capacity CAC—is here accurate enough for practical interpre- tations over the whole range of reported loadings. The same observation is applicable to the scenarios to be reported in Sub-
3
Peakedness of a cell stream is defined as the ratio of peak rate over average rate. Here, the peakedness is then equal to 7 .
0 0.5
1 1.5
2 x 1010
DAR With Peak-Rate CAC (Analytical) DAR With Equivalent-Capacity CAC (Analytical) Simulation ( 95% Confidence Interval )
0 10 20 30 40 50 60 70 80 90 100
Load Offered To Each Node Pair (Erlang) r0= 1
r0= 0 r0= 1
r0= 0
TotalMeanRevenueRate:(Units/s)W
Fig. 5. Smooth source scenario ( # $ " , ,
( ) #
* # # , ( ) #
. / ,
all in Mbits/s;2
#
* is varied;
peakedness " / ).
section V-B.
By setting , to 0 or 1, one obtains, respectively, the lower bound or upper bound of . Based on these bounds, Figs. 4 and 5 show that DAR with the equivalent-capacity CAC can return larger than DAR with the peak-rate CAC, especially for the peaky source scenario at high loads. This is because peaky sources allow, with the equivalent-capacity CAC, more economies of scale in statistical multiplexing of cell streams and the statistical multiplexing occurs at high loads.
For quantification, define thegain as
=< 4 " 4 < J <
4 < J < (9)
where =<4 4 <
J < denotes from DAR with the equivalent-capacity (peak-rate) CAC. For the peaky source sce- nario in Fig. 4, the maximum value of (found at 100-Erlang load) is approximately 450% and 205% when , is set to 0 and 1, respectively. For the smooth source scenario in Fig. 5, the maximum value of (found at 100-Erlang load) is approxi- mately 210% and 90% when , is set to 0 and 1, respectively.
B. Scenarios of Three Capacity-Guaranteed Services
This subsection considers two more scenarios each with three capacity-guaranteed services . The settings and results of both scenarios are depicted in Figs. 6 and 7.
In Fig. 6, the effect of link capacity is studied. It is observable from both the cases of , and , that the total mean revenue rates from DAR with the equivalent-capacity CAC and from DAR with the peak-rate CAC are equal when is very small as well as when is very large. The reason is that most of the calls are blocked, no matter what CAC is in use, when
0 50 100 150 200 250 300 350 400
0.5 1.0 1.5
TotalMeanRevenueRate:(Units/s)W
Capacity Of Each Link :C (Mbits/s) DAR With Peak-Rate CAC (Analytical) DAR With Equivalent-Capacity CAC (Analytical) Simulation ( 95% Confidence Interval ) x 1010
0
r0= 1 r0= 1
r0= 0
r0= 0
Fig. 6. Scenario of three capacity-guaranteed services ( / #
" $
, @ $ " , 3 / , is varied,
( ) * ( ) .
, all in Mbits/s;2 *
,$ ,/
).
is very small. Likewise, most of the calls are admitted, no matter what CAC is in use, when is very large. In between both the extreme values of link capacity, the equivalent-capacity CAC al- ways outperforms the peak-rate CAC in terms of the obtainable mean revenue rate (given the same setting of , at 0 or 1). In Fig. 6, is found to be maximized at about Mbits/s.
The maximum value of is approximately 100% for the case of , and approximately 50% for the case of , . This gain is relatively smaller than the gain obtainable from Figs. 4 and 5. This is because Figs. 6 and 7 also consider service 3, which is of a constant-bit-rate type ( ).
In Fig. 7, the effect of trunk reservation is examined. All directly routed calls are allowed the full-capacity access on ev- ery link and the parameters < : of alternatively routed calls are equally varied from 20 up to 60 Mbits/s. For comparison, in Fig. 7, the value of as obtained when all al- ternatively routed calls are allowed the full-capacity access is also depicted at < .
From Fig. 7, it is found that slightly increases and fi- nally saturates once < increases. An explanation is that, given the current setting, the network is heavily loaded. Ac- cordingly, once < increases, overflow traffics are suppressed and each link becomes more utilized by directly routed con- nections. For the same capacity-guaranteed service, a directly routed connection generally requires less link capacity than an alternatively routed connection because of two reasons. First, a direct route needs only one link while an alternative route needs two links. Second, a direct route generally contains more con- nections and those connections enjoy more economies of scale due to the nonlinearity in equivalent capacity. As a result, once overflow traffics are suppressed, the network can accommodate
