Analysis of the M and M 2Routings in Circuit-SwitchedNetworks
Eric W. M. Wong, Tak-Shing P. Yum and Kit-Man Chan
Department of Information Engineering
The Chinese University of Hong Kong, Hong Kong
need for detailed information passing between nodes and (3) no need for a
pre-planning of routingpatterns. TheDynamicollyContro~~edRouting
(OCR)
[41 proposed by Northern Telecom is a centralized routing rule. A central
routingprocessorreceives infomation every 10secondsfrom all the switches
and update their DCR tables accordingly. The choice of alternate routes is
based on the number of idle trunks and the exchanged utilization levels and
is therefore a state-dependentrule. More recently, a study was performed by
Ash, et. al[51 showing that it is feasible to monitor channel occupancies and
make routing decisions on a call-by-call basis.
Abstract
In nonhierarchical circuit-switched networks, calls can be routed to
alternate paths if the direct path is blocked. In this paper, we analyze two
alternate-path routing rules called the Madmum Free Circuit routing and the
Madmum Free Circuit with Minimm Occupied channel routing. For convenience, we shall call them the M and M2routings respectively. In the use
ofMrouting,acall isroutedtoanalternatepaththat has themaximum number
of free circuits when the direct path is blocked. The M2 routing is an
improvement of the M routing in that when multiple alternatepaths have the
samenumberof freecircuits, thepathwiththesmallesttotaloccupiedchannels
is chosen. Analytical results show that M2routing provides a small but significant improvement over M routing when the number of alternate paths is
large and/or the hunk group size is small. These results are verified by
simulation. As the impementationof Mzrouting is no more complicated than
M routing (both require the same channel occupancy information) and its
performance is always better than M routing, M2routing is deemed a better
d e to use.
Previous analytical studies in this area include the work of Krupp [6]
on Random Alternate routing with and without trunk reservation on symmetricalnetworks, the extensionby Akinpelu [7]on generalnon-symmetrical
networks and the incorporation of external blocking by Yum and Schwartz
PI.
In this paper, we analyze the performance of two stabdependent
routingprocedures on symmetricalfully connected networks. The f i i t one is
called MaximumFree Circuitrouting whose model, as reported in [9],is the
fist Fixed Point Model analyzed where the rate of the alternaterouted traffic
offered to an individual link depends on the state of the l i i . It directs an
overflowed call to an alternate path that has the maximum number of free
circuits. It was reported in [3,10] as the Least Busy Alternate routing. We
choose to call it Madmum Free Circuitroutingbecause it is more descriptive.
It will also not be confused with the second rule that we are studying in this
I. Introduction
Network management is "the supervision of the telecommunication
networktoassurethemaximumflowoftrafficunderallconditions"
[l].When
an overload occurs, various network management functions must be performed to control the flow of traffic to minimize network congestion. These
control functions include the reduction of operator traffic, recorded
announcements,altennate route cancellation, traffic rerouting etc. With the
use of common channel signaling and stored-program control, more sophisticatedcontrolfunctionscan be usedin networkmanagement. Among these
control functions,re-routing of traffic to less congested routes should always
be done f i i t , as it affects neither the customers nor the other network management functions.
papercalledMaximumFreeCircuitwithMinimumOccupiedChannelrouting.
We shall, for convenience,call the f i t one M routing and the second one M2
routing. M2routing is an improvement of M routing in that when multiple
alternatepaths have the samenumberof free circuits,the path with the smallest
total occupiedchannels is chosen. We shall show that the use of these mu
procedures together with trunk reservation can i
carrying capacity when compared to the use of
analytical difficulties, we shall use the same fullyuniformly loaded, nonhierarchicalnetwork model used in [6] and [8]. We
shall also use the same set of simplifyingassumptions in [6-81, namely that
the traffic statistics are assumed to be independent at each l i i and that the
alternately routed (or the overflowed) traffic is assumed to be Poisson.
In recent years, a variety of approaches to alternate routing networks
havebeen developed. AT&T has used adecentraliizednonhierarchicalrouting
strategy,called DynamicNonhierarchicalRouting (DNHR) [2] for a number
of years. DNHR is a he-dependent routing scheme that increases network
efficiency by taking advantageof the noncoincidence of busy hours in a large
toll network. 'Ihesecond approach,which is currentlybeing implemented in
theBritish Telecommain network, is calledDynamicAlternateRouting(DAR)
[3]. The DAR scheme has the advantages of (1) distributed control, (2) no
Recently, G m i a and Lockhart [111 applied Compartmental Modeling
to non-hierarchicalcommunicationsnetworks. This modeling is much more
complicated than ours, but it allows the formulation of network dynamics.
Our approachis to derive the steady stateperformance. More recently,Mitra,
1A87
n-78n3-0608-2/92/$3.00
o 1992 IEEE
AB has i busy channels, the total alternate-route traffic A, on link AB is
Gibbens and Huang [12] proposed a simplified implementation of the M
routing based on the aggregation of states. With proper design, it can substantially reduce signaling traffic with only a small loss of performance.
A, = 2rna,.
(5)
When links AB has channel occupancy i the call arrival rate h, and the
II. MRouting
call departure rate p are
We consider two cases here: without trunk reservation and with trunk
reservation.
A. Without Trunk Reservation
h, = D +Ai
i =0,1, ...,N - 1
ki = I
i = 1,2, ...,N
(6)
Consider a E node fully connected and uniformly loaded network where
Since the arrival rates are functions of the stateprobabilities, this ”birth-death’’
process can only be solved numerically by relaxation as follows. From (3,
we have
all links consist of N channels. Let P , be the probability that there are n calls
on a link (or that n channels are occupied). Then PN is the probability of
blocking on that link. Let D be the direct-route offered load to a link. Then
DPN is the overflowed load to the alternate paths. We shall restrict our choice
of alternate paths consisting of only two links. It was shown 1131 that the
total number m of such two-link alternate paths is equal to E - 2.
Consider a particular alternate path. Let the number of occupied
channelson thefiistlinkbeiand thaton the secondlinkbej. Then thenumber
of occupied circuits k in that path is k = max(i, j). When the direct path is
full, theM routing will direct the call to the alternate path with the maximum
number of free. circuits or with minimum k. When there are more than one
such paths, choose one at random.
N
which, through (l),can be expressed in terms of P i , P i t l ,...,PN. Next, the
balance equation for the above process says
(D +Ai)Pi= (i + l)Pi+l
k=o
I
i=O,l, ...,N-1.
(8)
Substituting (7) into (8), we arrive at a set of nonlinear equations. Let
i = N - 1, we obtain a nonlinesr equation with two unknowns PN and PN.
Assuming an initial value for PN say equal to PA0). Then P$! I can be solved
numerically. Repeated use of (8) with i = N - 2, i = N - 3,. .. allows us to
solve P${2,P$!a ...,P,,(O). Using the normalization equation Pio)can now be
updated as
-,
I1
a link has less than k
occupied channels
I’
i=o,1, ...N-1, (7)
J = l + l
Consider a particular path AC. If link AC is full, the overflowed AC
calls of rate DPNwill beroutedrandomly to one of the Maximum-Free-Circuit
paths (or M paths for short). Let there be a total of a such M paths. Then,
the alternate path load of AC that falls on a particular M path, say path ABC,
is DPN/a. Let Z, be the probability that a two link alternate path has k or more
occupied circuits. Then,
Zk= 1 -[€’rob[
)
=mpN[zy-zyi - , ? p, + N, -El zm-zjm+l
z,-zi+, 1 = , + 1
zj-zj+1
(9)
Repeat the above interations until certain accuracy criterion is met for PN.
The end-to-end blocking probability B, using M routing is therefore
Given that path ABC has k occupied circuits, the probability f ( a I k)
B, = Prob
that the a- 1other alternate paths also havek occupied circuits each and each
of the remaining rn - a alternate paths has more than k occupied circuits is
given by
on alld
[ Blocking ond [mBlocking
alternate path
the direct pa
(10)
= PN[I - (1 - PN)q
For the numerical results presented in section IV, a relative error of less than
lo4 was imposed on all end-to-end blocking probabilities.
whereZ, -Zk+lis theprobability that analternatepath haskoccupiedcircuits.
B. With Trunk Reservation
Therefore, given that path ABC has k occupied circuits, the amount of traffic
y(k) that gets routed from AC to alternate path ABC is
With trunk reservation, the last r free channels on a link are always
reserved for direct route traffic. Hence the call arrival and the departurerates
on a particular link become
(3)
ID
D+Ai
hi =
Therefore, given that link AB has i busy channels, the overflowed traffic ai
from link AC to link AB is
OSiSN-r-1
N-rSiSN-1,
pi = I
1
i = 1 , 2,...,N,
(110)
(1lb)
N-l
ai = J,E
y(max(i,j))Pj
=o
OSiSN-1.
(4)
where
N-r-1
Since link AB carries the alternate traffic from 2 m alternate paths, when link
Ai=
1488
C
. = n-
I
y(max(i,j))P,
i=O,l, ...,N-r-1.
(12)
For i > N - r ,we can solve the balance equation directly to obtain P , in terms
Therefore, given k and I , the amount of traffic y(k,l) that gets routed from AC
to alternate path ABC is
of P N - ,as follows:
N-r+l<i<N.
(13)
Therefore, substituting (13) into (12). A, can be expressed in terms of
= DPN
P,,P,+ I , . ..,PN-,. SubstitutingA, into the following balance equations
i = 0, 1 , ...,N - r - 1,
(D+ A,)P, = (i + 1)P,+ I
-zr+
1.1 + 1
+ zk + 1,I +
myk,f.
(19)
Therefore, given link AB has occupancy i , the overflowed traffic a, from link
AC to link AB is
(14)
(P,} can similarly be computed as in the last subsection. The end-to-end
blocking probability B,
(yk,f
N-l
a, = ,X y(max(i,j),min(i,jNPj
for M routing with Trunk Reservation is
i = 0, 1,
I = O
...,N - 1.
(20)
Sincel i AB carries the alternate traffic from 2m alternate paths, when
it has i busy channels, the total alternate-route traffic A, on it is
A, = h a , .
III. M zRouting
For Mzrouting, we also derive PNfor the two cases with and without
trunk reservation. The end-to-endblocking probabilities, denoted as B,, and
B,,, are given by (10) and (15) respectively with the new Pw
(21)
As before, the call arrival and the call d e p m rates are
A. Without Trunk Reservation
Consider one particular alternate path ABC of a direct path AC. Let
the number of busy circuits on the first and the second links be denoted as i
and j respectively. Then k = max(ij1 and 1 = min(ij7 are the occupancies of
the more busy and the less busy links respectively. When the direct path is
full, the M 2routing rule will route the call to the alternate path with minimum
k. When there are more than one such path, choose the one with minimum 1.
When there are more than one path with the same minimum k and minimum
I , choose one at random.
ki=D+Ai
i = 1,2, ...N ,
pi = i
i = O , l , ...,N - 1 .
(22)
To start the iterative solution of the state probabilities ( P i ] ,we observe that
N-1
Ai=h
,I:Pjy(max(i,j),mW,j))
1-0
Let Yk, be the probability that path ABC has k and 1 busy channels on
its two links. Then,
yk,f=
Let
i"
2P,P,
=')
k>l.
As before, substituting (23) into the balance equation, ( P i ) can be solved
recunsively as in the last section.
be the event that an alternate path has k or more occupied circuits
and c2bethe event that the alternate path has k - 1 occupied circuits and more
than 1 - 1 busy channels on the less busy link. As and are mutually
exclusive, Zk,,=Rob[ or <Jcan be computed as
B. With Trunk Reservation
cl c2
1
a link has less
Zk.,= 1 -prob than k occupied
channels
[ [
+
With trunk reservation, the last r free channels on a link are always
reserved for direct route traffic. Hence the call arrival and the call departure
rates on a particular link become
3
the busier link has k-1 occupied
h o b channels and the other link has
more than 1-1 occupied channel
[
b
D+A,
k,=
OSiSN-r-1
N - r S i I N - 1,
p,=i
I
i = 1 , 2,...,N ,
where
Moreover, given that alternate path ABC has k and 1 busy channels, let t3be
N-,-I
the event that there are a- 1 other alternate paths also having k and 1 busy
channels and 54 be the event that each of the remaining m -a alternate paths
has either more than k occupied circuits or has k occupied circuits and more
than 1 busy channels on the less busy link. Then,f(a I k,I ) = R o b [<, and
is given by
A, = 2m
,E Pjy(max(i,j),min(i,j ) )
i = 0, 1, ...,N - r - 1.(25)
J = Q
a
For i > N - r ,we can solve the balance equation directly to obtain
1489
+,,...,PN-,.
Substituting(26) into (25), A, can be expressed in terms of P,,
P,
Further substituting into the balance equation, [Pa)can again be solved
recursively.
V. Conclusions
We have analyzed the M and M' routings using a fiied point model
where the rate of the alternate traffic offered to a link depends on the state of
the link. The M' routing is found to provide a small but significant
improvement over M routing when the number of alternate paths is large
and/or the trunk group size is small. As the implementation of M2 routing is
no more complicated than M routing (bothing requiring the same channel
occupancy information)and its performance is always better than M routing,
M2 routing is deemed a better rule to use.
IV. Performance Comparisons
Figure 1shows the stationary state probability of M 2routing with trunk
reservation under differentdirect trafficloading. The truncated Gaussian form
of the stationary state probability distribution is observed. As direct traffic
increases,the dump bell curve shiftsto the right, yielding a larger end-to-end
blocking probability.
We have also studied the performance of the reversed M2 routing, i.e.,
the rule that chooses an alternate path with minimum occupancy fist, and if
thereisatie,chooseonewiththemaximumnumberoffTeecircuits. Extensive
simulation on a 9-node fully connected symmetric network shows that the
end-to-end blocking probability is vitually the same as that for M' routing
under moderate to heavy traffic conditions. More study is needed to explain
why this is so. Other state-dependentrules can be formulated with different
uses of the channel occupancy information and more eleborate routing rules
should also take the traffic rates into consideration.
Figure 2 shows the alternate traffic rate of Random Alternate Routing
(RAR) and M' routing with trunk reservation as a function of states for D
equals 85,90 and 95. We observe a sharp drop of A, at a certain state and this
drop becomes sharper as D increases. Comparing M2 with RAR, we see that
at moderate traffic load (say D=85) M' routing has higher alternate traffic at
lower states and smaller alternate traffic at higher states. This fact reflects
the ability of M' to route alternate traffic to less congested alternate paths.
We also observe that the distributionof alternate traffic rate of M2 get closer
to that of RAR as D increases. This shows that in heavy traffic conditions,
the improvement on blocking of M' over that of RAR is not as significant as
compared to that in moderate traffic conditions.
References
Figure 3 shows the simulation results (those with markers) and the
analyticresults of the blocking probabilitiesof M' routing for various r values
with N=30 and D=27. It is found that the analytic results match very well
with the simulation results except for PO, where the results are a little bit off.
Similar behavior was found for RAR, a plausible explanation was given in
Wl.
Figure4 showsthe percentageimprovementon the end-to-end blocking
probabilityofM2routingoverthatofMrouting asafunctionofD withN=lO
and m=6. The M' routing has a property that its relative improvement over
its counterpart depends on the direct traffic rate. A maximum of 30% and
16% relative improvements on the end-to-end blocking probability are
observed for the case without and with trunk reservation.
Figure 5 shows that for D=3 and N=5, the end-to-endblocking prob
ability given by M' routing without trunk reservation is always smaller than
that of M routing, independent of the network size. Similarbehavior is found
for other combination of D and N, and for the case with trunk reservation. It
is also observed that without trunk reservation, the blocking increases with
the number of alternate paths. This is also shown in Figure 3 when r=O, ie,
comparing B for m=l and m=8.
Figure 6 shows the end-to-end blocking probability of M2 routing
against direct traffic load for different number of alternatepath using optimal
trunk reservation parameters. Table 1 shows the optimal r values. It is seen
that the optimal r increase with D and M. This figure shows that with the use
of optimal r, the blocking probability decreases with increasing m. Hence,
all availablealternatepaths in a network should be used provided that optimal
r is also used
1.
J.G. Pearce, TelecommunicationsSwitching, Plenum Press, New York,
1981, pp. 301-304.
2.
G.R. Ash, R.H. Cardwell and R.P. Murray, "Design and Optimization
of Networks with Dynamic Routing," BSTJ, vol. 60, pp. 1787-1820,
October, 1981.
3.
RJ. Gibbens, "Dynamic Routing in Circuit-Switched Networks: The
Dynamic Altemate Routing Strategy," PhD thesis, University of
Cambridge, 1988.
4.
W.H. Cameron, I. Regnier, P. Galloy and A.M. Savoie, "Dynamic
Routing for Intercity Telephone Networks," ITC-10, Montreal, 1983.
5.
G.R. Ash, J.-S. Chen, A.E. Frey and B.D.Huang, "Real-TimeNetwork
Routing in an IntegratedNetwork, submitted to ITC 13, Copenhagen,
1991.
6.
R.S. Krupp, "Stabilization of alternate routing network," IEEE Int.
Commun. Conf.,Philadelphia, PA, June 1982, pp. 31.2.1-31.2.5.
7.
J.M. Akinpelu, "The overload performanceof engineered networks with
nonhierarchicaland hierarchicalrouting," Bell Syst. Tech. J., vol. 63,
no.7, pp.1261-1281,1984.
8.
T.K.Yum and M. Schwartz, "Comparison of routing procedures for
circuit-switched traffic in nonhierarchical networks", IEEE Trans. on
Commun., vol. COM-35, no. 5, pp. 535-544,1987.
9.
Eric W.M. Wong and T.3. Yum, "Maximum Free Circuit Routing in
Circuit-SwitchedNetworks', Proc. IEEE INFOCOM'90, IEEE Computer Society Press, pp. 934-937,1990.
10.
W.Y.Ng, "DynamicRouting in Circuit-SwitchedNetworks:Simulation
of the Routing Rule of Least Busy Alternate with Trunk Reservation,"
Part I1 Project, Dept. of Engineering,University of Cambridge, 1985.
11.
M.R. G m i a and C.M. Lockhart, "Nonhierarchical Communications
Networks: An Application of CompartmentalModeling," IEEE Trans.
on Commun., vol. COM-37, no. 6,pp. 555-564, June 1989.
Figure 7 shows the percentageimprovementof M'over M routing with
trunkreservation for different values of r whereN=10, D=20/3. We observed
that the percentage improvement on blocking probability increases with M.
This phenomenon is not found in the case without trunk reservation (c.f. Fig.
5).
1490
D. Mitra. RJ. Gibbens and B.D.Huang, "State-DependentRouting on
Symmetric Loss Networks with Trunk Reservations, I", to appear in
IEEE Trans.on Communications.
12.
13.
Alternate traffic rate
50
M. Schwartz, TelecommunicarionNetwork: Protocols, Modeling and
M2, D=9O
Analysis, Addison Wesley, 1988, pp.334.
RAR, D=95
40
Table 1 : Optimal trunk reservation parameters
30
RAR. D=9O
20
M2. D=85
10
RAR. D=85
0
0
10
20
40
50
60
channel states
30
70
80
90
100
Fig. 2
Alternate traffic r a t e vs s t a t e s
for M 2 routing and RAR with TR
0.3
Stationary state probability
0.09
D=
0.075
0.06
i
1
0.25
N=100
m=30
0.2
D=85
r
B
0.045
0.15
\
0.03
0.1
0.015
0.05
0
0
10
20
30
40
50 60 70
channel states
80
90
100
I
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
I
I
9 1 0 1 1
Y
Fig. 3
Blocking as a f u n c t i o n of
TR p a r a m e t e r s , M2 r o u t i n g
Fig. 1
S t a t i o n a r y s t a t e probability
of M2 r o u t i n g with TR
1491
35 [ercentage improvement
Call blocking probability
0.22
0.2
0.18
0.16
0.14
4
5
6
7
8
9
1
0
D
0.12
(a) Without trunk reservation
0.1
Percentage improvement(%)
20
0.08
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10
D
Fig. 6
Blocking p r o b a b i l i t y of M 2
with optimal r
Percentace improvement
35
F
4
5
6
7
8
9
10
D
30
N=10
(b) With trunk reservation
Fig. 4
P e r c e n t a g e i m p r o v e m e n t of
M2 over M
0.08
25
3
20
15
10
5
0
2
3
4
5
6
7
8
9 1 0 1 1 1 2 1 3 1 4
m
01
1
"
3
5
"
7
9
I1
'
'
'
'
13
15
17
19
Fig. 7
P e r c e n t a g e i m p r o v e m e n t in B ,
M 2 over M w i t h TR
m
Fig. 5
Blocking c o m p a r i s o n of
M 2 and M w i t h o u t TR
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