a Routing Algorithm based on QoS Estimation and

TBPE: a Routing Algorithm based on QoS Estimation and Forecast Vincenzo Eramo, Ileana Martino, Ugo Mocci, Caterina Scoglio Fondazione Ugo Bordoni, via B. Castiglione 59, 00142 Roma, Italy e-mail:[email protected]

Abstract In this paper we propose the TBPE - Transient Blocking Probability Estimation routing algorithm, an adaptive method based on Grade of Service forecast requiring limited network monitoring and computational complexity. This routing scheme is compared with the well-known routing algorithm class MDP in which Markovian Decision Process theory is used for the QoS forecast. Both routing algorithms are tested in terms of robustness and flexibility, that are key features in the future networks, due to variability and uncertainty of traffic. The numerical results show that the performances of TBPE and MDP are similar but they differ for the monitoring effort and the computational complexity. 1. Introduction Future multiservice networks will operate with very heterogeneous, variable and uncertain offered traffic. In this environment, new techniques have to be developed in order to meet three kinds of objectives: 1. usage efficiency; 2. QoS fairness; 3. robustness. In particular, Connection Admission Control (CAC) and Traffic Routing Algorithms have to be suitably designed to reach such objectives. Many traffic routing techniques have been proposed and implemented. In particular dynamic “state-dependent” routing schemes were introduced in order to adjust the routing paths in accordance with the variations of the offered traffic. Recently “state-dependent” routing and CAC schemes based on the forecast of a performance index have been proposed in the literature [Hwa1, Kri1, Kri2, Zha, Ray, Dzi1, Dzi2]. In these algorithms the decision “if” accept the connection and eventually “where” to route it, is made evaluating the “revenue” and the “cost” of the decision. The revenue is directly related to the connection acceptance and the resource usage. It depends mainly on the characteristics of the connection and not on the network state. Instead, the cost represents the average revenue lost in the future due to the connection acceptance; it depends on the connection characteristics, the network state and the forecast process. If the traffics are poissonian, this last process

Work carried out in the framework of the agreement between Telecom Italia and Fondazione Ugo Bordoni

V. Eramo, I. Martino, U. Mocci, C. Scoglio

TBPE: a Routing Algorithm based on QoS Estimation and Forecast

can be mathematically described by a Markovian Decision Process (MDP) and the routing algorithm is called MDP routing [Hwa2]. The MDP routing is described in section 3 under simplified assumptions, such as the “link decomposition” and the “link occupation” aggregate state variable, which reduce the dimensions of the problem, making it solvable and feasible. This method requires a high network monitoring rate, since the routing decision is taken call by call on the basis of the measured network state. To overcome this limitation still maintaining the performances, we propose a new routing method (TBPE routing) which requires a periodic (with period T) network monitoring, where T is greater than the mean call interarrival time. This method is described in section 3. As the primary goal of the paper concerns the robustness, in section 4 the robustness concept is introduced, defining a set of significant overload conditions and a proper index of robustness. In section 5 TBPE routing is tested on a logical full connected network. Performance, complexity and robustness of the method are evaluated and compared with other methods, i.e., MDP, LBA (Least Busy Alternative Path), SSO (Static Sequential Overflow). 2. Description of the MDP algorithm To illustrate the MDP routing algorithm we consider k (k = 1, 2, …, D) traffic classes, with class k described by {λk, µk, rk}, where λk is the Poisson arrival rate, 1/µk is the mean holding time, rk is the call bandwidth requirement. The MDP routing algorithm [Hwa2] is described by the flow diagram on figure 1a. A call between the nodes (m, n) belonging to the traffic class k, is accommodated on the single-link direct route if there is enough bandwidth and if the cost of the link (m, n) involved by the call, Ck(sm,n), depending on the state sm,n, is greater than the revenue α k, which is independent of sm,n. The state sm,n=(s 1m,n, s 2m,n, ..., s D m,n) is a vector whose components are given by the number of calls active on the link for each traffic class. Ck(sm,n) is an estimate of the gain missed in the future due to the call acceptance and the resource occupation. If the call is rejected on the direct link (m, n) the algorithm computes the cost of each other multi-link route admissible for the call. The cost of each route is the sum of the costs due to the admittance of the call on the links of the route. Let V be the cost of a minimum cost route, then the call is accepted on such route if V is smaller than αk, otherwise it is rejected. The problem of computing the costs Ck(sm,n) can be set as a Markov decision problem [How1, How2]. Omitting for simplicity the subscripts m, n, we denote with S the set of the admissible states of a link with capacity C, i.e.: S ={s / ∑

D i=1

si ri ≤ C}



The costs assume the expression Ck(s) = ν (s+k) - ν (s) where s+k = (s1, ..., sk+1, ..., sD) that is the state assumed by the link if the call is accepted. ν (s) are the Howard’s relative values [How1, How2] of the states with respect to some fixed state, e.g., the zero-occupancy state (s1 = 0, ..., sk = 0, ..., sD = 0).

ν (s) is governed by the following linear equations system: L = x(s) + ∑p∈S a(s, p) ν (p)

s∈ S

(2.1)

where: –

L ≡ loss rate L=∑

D

k=1α kλ kp k

with pk the equilibrium loss probability for the traffic class k; –

x(s) ≡ loss rate in the state s, i.e., the sum of arrival rates of the streams blocked in the state s; x(s) = ∑

D

k=1α kλ ku k(s )

with: 1 uk (s) =  0 –

if

s+k ∉ S

if

s+k ∈ S

a(s, p) is the transition rate from the state s to the state p of the Markovian process compatible with the link occupancy states; The computation of ν (s) is very hard due to the dimension of the system (2.1) that can be very high.

In [Kri1] it is proposed to represent the link by means of a scalar variable i (i = 0, 1, 2, ..., D) indicating the overall bandwidth occupied on the link. Therefore we introduce sets of “aggregated” scalar states of the original Markov chain. According to [Kri1, Kau1, Kau2] the obtained process can be approximated with a Markov process whose transition rates have the following expression:



ai,i+rk = λk

for

i ≤ C-rk

k = 1, 2, …, D

ai,i-rk = λk q(i - rk) / q(i)

for

i ≥ rk

k = 1, 2, …, D

ai,j=0

for

ai,i = −

all other j (j ≠ i)

(2.2)

C

∑

ai,u u=0 u≠i

where q(i) is the equilibrium probability that i bandwidth unit are occupied on the link. We can use Kaufman’s one-dimensional recursion formula to determine the probabilities q(i). Therefore the costs are determined by the expression C K(s) =ν (s+k) - ν (s) ≈ νi - νj, where: –

i=∑

D

p=1 rp s

+k,p,

j=∑

D

p,

p=1 rp s

being s+k,p the p-th component of the state vector s+k;

– ν i determined by the following linear equation system: L = xi + ∑

C j=1

ai,jνj

i = 0, 1, ..., C

where xi is given by the relations xi=∑

D p=1

λpαpup (i)

0 u p (i ) =  1

if i + rk ≤ C if i + rk > C

It proved in [Kri1, Kau1, Kau2] that the introduced approximations are expected to be adequate for appreciable levels of traffic loads and link bandwidths. In such situations, an admission control will determine, for each level of link occupancy, the traffic classes to be admitted and the ones to be refused. This characteristic allows us to formulate a simple connection acceptance policy, based just on knowledge of the total occupancy of a link (without considering the precise mix of calls in progress). The simpler criterion, of course, is not as precise as the one based on the exact state space, but is significantly easier to implement. 3. Description of the TBPE algorithm To reduce the complexity of the MDP algorithm, we introduce a new routing algorithm (TBPE) where the network is monitored every T sec. The TBPE algorithm is described by the flow diagram on figure 1b. A call between the nodes (m, n) belonging to the traffic class k is accepted on the direct single-link route if there is sufficient bandwidth. If the call is rejected the algorithm computes the average blocking probability on the interval T of each



admissible multi-link route. This probability is evaluated by assuming statistical independence between the call losses on each link of the route and the call is routed on the minimum average blocking probability route under the condition that the average blocking probability on each link of the route is smaller than a threshold TRk so to protect the direct traffic from the overflow traffic. The monitoring interval T is chosen as trade-off between the estimation effectiveness and the reduction of the monitoring effort. The average blocking probability on the interval [th, th + T] for the calls belonging to the class k, Pk[sm,n(th)], is obtained by the following expression: Pk[sm,n(th)] = (1/T) ∫th

th + T

Pk[sm,n(th), t] dt

where: – sm,n(th) is the state of the link at instant th; – Pk[sm,n(th), t] is the blocking probability at instant t in the case the system is in the state sm,n(th) at instant th. We can write: Pk[sm,n(th), t] = ∑

C l=C-rk+1

ql(t)

where qi(t) is the probability that i bandwidth units are occupied at instant t. These quantities are calculated by solving the transient of the Markov Chain of the sets of “aggregated” scalar states introduced in the section 2. Therefore the vector q(t)=[q1(t), q2(t), ..., qc(t)]T is given by the following differential equations system: dq(t ) = AT q(t) dt

t ∈ [th , th + T].

(3.1)

The solution of (3.1) is given by q(t) = eAt q(th) where: – A is the transition rates matrix of the mentioned Markov chain, whose elements are given by (2.2); – q(th) is the initial condition vector of the system. If i bandwidth units are occupied at instant th, q(th) is given by q(th)T = [q 0 = 0, ..., q i-1 = 0, q i = 1, q i+1 = 0, ..., q C = 0]T. We set the threshold TRk according to the relation:



   r µ  TRk = 1 − n k k  p*  ∑ ri µi    i =1

(3.2)

where the parameter p* can be determined by means of simulations. From (3.2) we can see that on the overflow route the calls with long average time and requiring more bandwidth will be rejected in order to route the calls requiring less resources. 4. Robustness evaluation As robustness is a rather wide concept a quantitative analysis can be carried out only after a precise definition. We define robustness of a routing scheme as the capability to maintain the total carried traffic above a given threshold with respect to the maximum in a given set of significant “overload” conditions. In a context where connection requests arrive according to a Poisson process and the offered traffic is O, the carried traffic is bounded by the Erlang Bound carried traffic TE(O) [GK]. The relevant Erlang Bound blocking probability represents a lower-bound of the connection blocking probability for any dynamic routing scheme in the mentioned stationary condition. The definition of a set of significant overload conditions is a prerequisite for the robustness evaluation, because there exist infinite offered traffic matrices which represent a traffic variation with respect to the nominal traffic Oref. For analysis purposes we focus on the following three classes of overloads: – Relation Overload RO(x, r): traffic offered to the traffic relation r is incremented by x ; – Node Overload NO(x, n): traffic offered to each relation from/to the node n is incremented by x; – General Overload GO(x): traffic offered to each relation is incremented by x; These definitions have been applied to the sample network of figure 2. In each overload class the different overload conditions are ordered according to the increasing offered traffic O, which can represent the traffic offered to the whole network or to a set of network elements. After defining the reference and the overload traffic, we can give the following definition of robustness of a routing scheme: the robustness I(RS) of the routing scheme RS with respect to a given overload class and a discrete set of increasing offered traffic O∈[O ref,..,O N] is the maximum offered traffic Omax (Oref ≤ Omax ≤ ON) normalised with respect to Oref such that the corresponding carried traffic T(Omax) is not less than a given fraction f (0 ≤ f ≤ 1) of the Erlang Bound carried traffic TE. More formally: I(RS) = [(Omax – Oref) / Oref] × 100 where:



Omax : ∀Ο , Oref ≤ O ≤ Omax T(O) ≥ fTE If the carried traffic is always greater than the given fraction f ( 0 ≤ f ≤ 1) of the Erlang Bound carried traffic, we have I(RS) = UP, i. e. the upper bound of the considered range. 5. Numerical analysis In this section we compare each other the considered routing algorithms, MDP and TBPE, and two other routing schemes, LBA (Least Busy Alternative Path) [Kol1, Kol2, Cha] and SSO (Static Sequential Overflow), from the viewpoint of their robustness. To this purpose we consider the fully-connected fournode network of figure 2. A trunk reservation threshold [Son], tr, is adopted to protect fresh traffic from overflow traffic in shared resources. In both LBA and SSO tr is fixed and equal to few capacity units (c.u.), since this value allows good performance in a wide range of traffic conditions, as proved in [GK]. Concerning TBPE, the protection of fresh traffic is realised indirectly fixing the parameter p* introduced in section 3, which is a function of the monitoring interval. The numerical evaluations have been carried out in two parts considering first a single traffic network, and then extending the analysis to a multiservice network. All the computation has been obtained by simulation (Montecarlo method). Single service network Running these simulations we have assumed that the call bandwidth is equal to one c.u. and the network links have a capacity of 100 c.u.. The reference offered traffic Oref is 540 Erlang (90 Erlang for each traffic relation); further traffic patterns, belonging to the overload classes described in section 4, have been offered to the network. The variation of p* with the monitoring interval T and the corresponding blocking probability obtained with TBPE are shown in figure 3. For T=1/30 sec the optimal value of p* is equal to 0.001, while for T=1 sec p*=0.026. Increasing T 30 times the minimal blocking probability increases by about 20%. In the numerical analysis we have chosen: T=1 sec and p*=0.026. With General Overload GO(x) (figure 4) the maximum carried traffic is achieved with both MDP and TBPE; LBA and SSO have similar performances when the same value of tr is adopted and the carried traffic with tr = 2 is greater than the carried traffic with tr = 0. Similar results are also obtained in all the other different overload conditions considered in section 4. The robustness index I is also evaluated in figure 4 fixing f=0.9. In the considered range of overload conditions TBPE, MDP, LBA tr=2 and SSO tr=2 are always above the 90% Erlang traffic threshold, giving I(TBPE)=I(MDP)=I(LBA tr=2)=I(SSO tr=2)=UP. On the contrary with LBA tr=0 and SSO tr=0, when the offered traffic increases beyond 560 Erlang, the carried traffic start to be less than the 90% Erlang traffic and I(LBA tr=0)=I(SSO tr=0)=4%.



A similar behaviour is found also in case of Node Overloads NO(7%, 1), NO(10%, 1), NO(17%, 1), NO(33%, 1) and NO(50%, 1) (figure 5), where we find I(TBPE)=I(MDP)=I(LBA tr=2)=I(SSO tr=2)=UP and I(LBA tr=0)=I(SSO tr=0)=7%. In case of Relation Overloads RO(7%, a), RO(10%, a), RO(17%, a), RO(33%, a) and RO(50%, a) (figure 6), we find I(TBPE)=I(MDP)=I(LBA tr=2)=I(SSO tr=2)=UP and I(LBA tr=0)=I(SSO tr=0)=22%. Multiservice network In this last set of simulations we assume that two traffic classes K1, K 2 are offered to the network: the first class offers calls requiring 1 c.u., the second one 3 c.u.. The network links have a capacity of 150 c.u.. The reference offered traffic Oref is equal to 840 Erlang × c.u. (20 Erlang / traffic relation for K 1 , 40 Erlang / traffic relation for K2). Even in this case further traffic patterns, belonging to the overload classes described in section 4, have been offered to the network. Concerning the trunk reservation, it can be also used in a multiservice network to distribute QoS among the traffic classes; however, in our analysis we consider trunk reservation only to protect fresh traffic from overflow traffic, as in the case of single service network. Concerning the threshold tr we have chosen tr=6, T=1 sec and p*=0.1, for which the blocking probability is minimal (figure 7). In the different types of overloads (figures 8, 9, 10) we have also reported the results for LBA and SSO with tr=0. The best routing schemes are still MDP and TBPE, while the worst performances are obtained by LBA and SSO with tr=0. The traffic carried with LBA and SSO when tr=6 is just 1% less than the carried traffic with MDP and TBPE. Similar results are also obtained in all the reported overload conditions. Similarly to the single service network, we can compute the robustness by the index I introduced in section 4. For General Overload conditions (GO(20%), GO(30%), GO(35%), GO(40%), GO(50%)) and fixing f=0.9 (figure 8), TBPE, MDP, LBA tr=6 and SSO tr=6 are always above the 90% Erlang traffic threshold, giving I(TBPE)=I(MDP)=I(LBA tr=6)=I(SSO tr=6)=UP. On the contrary with LBA tr=0 and SSO tr=0, when the offered traffic is lightly greater than the reference offered traffic, the carried traffic start to be less than the 90% Erlang traffic threshold, so that I(LBA tr=0)=I(SSO tr=0)≅0%. A similar behaviour is found also in case of Node Overloads NO(20%, 1), NO(30%, 1), NO(35%, 1), NO(40%, 1) and NO(50%, 1) (figure 9) and in case of Relation Overloads RO(20%, a), RO(30%, a), RO(35%, a), RO(40%, a) and RO(50%, a) (figure 10).



6. Conclusions In this paper we have proposed a new routing algorithm TBPE based on network state forecast. At the beginning of each monitoring interval T, link occupancy is measured and the average link blocking probability estimated resorting to a transient time analysis; during the whole T interval, calls, rejected on the direct path, are routed on the minimum estimated blocking probability path. TBPE is simpler than the MDP routing algorithms founded on the Markov decision processes theory, which also use forecasts but require network monitoring call-by-call. In the considered network overload conditions the robustness performances reached by TBPE are as good as the ones obtained by MDP. Furthermore, we show that some other classic routing schemes, LBA and SSO, provide reduced performances in comparison with the ones obtained by TBPE. Finally the results prove a good robustness of TBPE in the range of the considered overload conditions, as the classic mentioned routing methods when provided with a trunk reservation mechanism.



References [Cha] [Dzi1] [Dzi2] [GK] [How1] [How2] [Hwa1] [Hwa2] [Kau1] [Kau2] [Kol1] [Kol2] [Kri1] [Kri2] [Ray] [Son] [Zha]

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ARRIVING CALL BELONGING TO THE CLASS k

ARRIVING CALL BELONGING TO THE CLASS k

EVALUATION OF COST Ck(Sm,n) DUE TO THE ACCEPTANCE ON THE DIRECT LINK

EVALUATION OF AVAILABLE BANDWIDTH ON THE DIRECT ROUTE

YES

Ck(Sm,n) < k

IS THERE AVAILABLE BANDWIDTH?

CALL ACCEPTED ON THE DIRECT LINK

EVALUATION OF COST DUE TO THE ACCEPTANCE ON THE ALTERNATIVE ROUTES Vp=Ck(Sm,p)+Ck(Sp,n)

EVALUATION OF COST ON THE ALTERNATIVE ROUTES

FIND A TWO LINK ROUTE HAVING MINIMUM COST

FIND A TWO LINK ROUTE HAVING MINIMUM COST V

Ck(Sm,p) < TRk

YES