A novel algorithm for dynamic admission control of elastic flows

Transcription

1 A novel algorithm for dynamic admission control of elastic flows Franco Blanchini, Daniele Casagrande and Pier Luca Montessoro Abstract The task of assigning part of the forwarding capability of a router to different flows, usually called admission control, is considered and an algorithm to handle the requests is developed. The idea is to admit not only the acceptance and the rejection answers but also a third kind of answer that occurs when there is no available share of the resource at the instant of the request but there may be a quote of resource available in a determined time-window in the future, provided that the active flows are suitably decreased. The algorithm guarantees both the fair occupancy of the resource and the optimality of its usage. Moreover, the only information on which the algorithm relies is the number of flows, and for each one, the minimum bandwidth needed and the desired bandwidth. Index terms Bandwidth assignment, admission control, elastic flow. I. INTRODUCTION When a resource is shared among several users, managing the allocation of the resource to the users requesting becomes adifficult task if fairness among users and the maximization of the usage are to be guaranteed. In this article we consider the task, performed by a router, of assigning part of its forwarding capability to different flows, usually called admission control, and we develop an algorithm to handle the requests. In the literature, diverse algorithms are available for admission control [1], [2], [3], [4], [5], [6]. However, to the best of these authors knowledge, all the available methods perform a straight rejection, i.e. either the router admits a new flow and assign to it some rate of transmission, or the request is rejected. On the contrary, the idea developed herein is to admit also a third kind of answer when there is no available share of the resource at the instant of the request but there may be a quote of resource available in a determined time-window in the future, provided that the active flows are suitably decreased. In this way a user, the request of which is unsatisfied, can decide to wait and try again after a given time-interval. The algorithm described herein guarantees both the fair occupancy of the resource and the optimality of its usage. The control works on a per-flow dynamic resource reservation schema, made scalable by the underlying algorithm REBOOK [7], that provides constant cost for flow table access regardless the number of active flows. Moreover, another advantage of the proposed algorithm is that the only information on which it relies consists is the number of flows, and for each one, the minimum quote, namely the minimum bandwidth needed for the source to be able to deliver the service, and the desired quote, namely the bandwidth requested by the source for optimal quality level. The task of the router is modeled making use of two different discrete time scales. The first, with a smaller time period, corresponds to the time scale of the communication between the router and the end nodes; this is also the time scale of the upgrading of the flow information table. The second time scale, with a larger time period, corresponds to the action of the control law. The event of a new source requesting the use of the channel and the event of a source stopping using it are treated asynchronously with respect to the time scale of the control law and, more importantly, no assumption is made on the statistics of the requests; hence, the validity of the method is general. II. NOTATION The following notation is used throughout the article. n(t) is the total number of sources transmitting to the router at time t; l i [, 1], i = 1,...,n(t), denotes the minimum level of transmission rate of the i-th source, namely the transmission rate below which the source cannot transmit; r i [, 1] denotes the transmission rate requested by the i-th source; obviously r i l i ; x i (t) [,r i l i ] is the quote, namely the amount of transmission rate at time t, associated with the i-th source, that can be assigned managed by the controller; R i (t) =x i (t)+l i denotes the total transmission rate assigned at the instant t to the i-th source; R i (t) [l i,r i ]; y(t) denotes the overall transmission rate at time t, thatis n(t) y(t) = R i (t). We assume that all the quantities are normalized, namely that the capacity of the channel is 1. III. STEADY STATE In this section we describe a model for the dynamics of the manageable amounts of transmission rate x i in the simplified scenario in which no new request occur. Obviously, this is not the final purpose of this work; however, this first result is accessory for the theory that is developed in the remainder of the article. To begin with, we make the ideal assumption that all the quantities vary continuously with time and we suppose that the router can behave as a control system, namely that it can affect the time-variation of the rate of transmission assigned to each source, thus obtaining the following model: ẋ i (t) =u i (t), (1) Page 11

2 where u i (t) is the input (decided by the controller). If the number of sources using the channel is constant, namely that n(t) =n for all t, then the following result holds. Proposition 3.1: If there exist x i,fori =1,...,n,such that x i [,r i l i ] and n ( x i + l i ) 1, then the input u i (t) =α( x i x i (t)), (2) where α is a positive constant, is such that x i is an equilibrium for Equation (1) and the equilibrium is exponentially stable. Proof It is sufficient to consider the error variable e i (t) x i x i (t). Its dynamics are ė i (t) = αe i (t), what implies that e(t) tends to zero exponentially. We mentioned in the Introduction that according to the model considered herein, the router has the information about the minimum transmission rate l i and the requested transmission rate r i associated with each flow; a reasonable choice for the steady-state values of the transmission rates may be based on these data. In particular, we define the steady-state fair distribution factor (SSFDF) F SS = and we choose 1 if n r i < 1, 1 n l i n (r i l i ), otherwise, (3) x i =(r i l i )F SS. (4) The choice (4) guarantees a fair share of the resource among the users. However, the maximization of the resource usage is also guaranteed, as the following result show. Proposition 3.2: The control law (2) with x i given by (4) is such that the occupancy of the channel is optimized. Proof. The time-variation of the occupancy of the channel is ẏ(t) = n n ẋ i = α ( x i x i (t)) = = α ( F SS n (r i l i ) ) n x i (t). Two cases are possible. If n r i < 1, thenf SS =1, x i = r i l i and the steady state value of y, i.e. the occupancy of the channel at the equilibrium, is ȳ = n (l i + x i )= n r i, which means that each flow is assigned its maximum request. Otherwise, if n r i 1, then we obtain ( ) n n ẏ(t) =α 1 l i x i (t) and the equilibrium value for y is ȳ = n (l i + x i )= n l i + F = n (r i l i )= ( n l i + 1 namely the full occupancy of the channel. ) n l i =1, The control law considered in Proposition 3.2 is different for each flow and depends, for the i-th flow, on the equilibrium values x i, namely on r i and l i, which are an information needed by the router to compute the SSFDF. Moreover, the router has to communicate to the sources the value of the current SSFDF, which is constant in the simplified scenario considered in this section but, in a real-world application, varies in time. The computation and the transmission to the active sources of the SSFDF at any time instant can be avoided by using a different control law, based only on the number of sources, n, and on the occupancy of the channel, y(t), as shown by the following result. Proposition 3.3: Let { n } ȳ min r i, 1. (5) The input {, if xi (t) =r u i (t) = i l i, (6) α(ȳ y(t))/n, otherwise, where α is a positive constant, is such that the occupancy of the channel is optimized. Proof. With the control laws (6) the dynamic equation for the total occupancy of the channel is n n ẏ(t) = Ṙ i (t) = ẋ i (t) = α (ȳ y(t)), (7) n i L(t) where L(t)={i : x i (t) r i l i }. Hence there are two stable conditions, namely L(t) =, meaning that each source is assigned its maximal quote, or y(t) =ȳ. In turn, the latter case can be split into two sub-cases; if ȳ =1 the channel is fully occupied; if ȳ 1 each source is assigned its maximal quote. Remark 3.1: By (5), we always have ȳ y(t). This property, in turn, implies that u i (t) is always positive, namely that x i (t) reaches its equilibrium value increasing monotonically. The model (1)-(2) does not consider that other sources may request the usage of the channel. To take into account this occurrence we may operate as follows. Suppose that a new source, described by parameters l n+1 and r n+1, requests to use the channel at time τ. Then three case are possible. Refusal. If l n+1 is such that n+1 l i > 1, then there exists no condition in which the n +1 sources can all use the resource; the request must be refused. Page 111

3 Acceptance. If y(τ)+l n+1 1, then the new source can be assigned a rate of transmission R n+1 such that l n+1 R n+1 1 y(τ); this can be done without affecting the transmission rate assigned to the other users. In principle, any criterion could be used to choose R n+1 ; however, to keep the distribution of the bandwidth fair we introduce an additional variable, named the dynamic fair assignment factor (DFAF), denoted by F DA, that is updated at any instant in the timescale of the controller, and we set R n+1 =min{1 y(τ),l n+1 + F DA (r n+1 l n+1 }. Delayed request. The third case is when n+1 l i < 1, which means that in principle the n +1 users could all use the channel at the same time, but y(τ) +l n+1 > 1, which means that the active n users at time τ are not leaving enough resource share for the n +1-th user. In this case the router needs to design a strategy in order to reduce (at least some of) the quotes x i,fori =1,...,n, in order to meet, after T time units, the condition y(τ + T )+l n+1 1. As a consequence, the n +1-th source cannot use the channel at time τ but will be allowed to use it at time τ + T. Its request is delayed of T time units. We describe the third occurrence more into details in the following section. IV. NEW REQUESTS AND DROPPINGS To properly model the scenario in which the router is endowed with a controller, we suppose that the time scale of the router is continuous while the controller performs its action periodically every T C time-units, thus its dynamics is modeled in a discrete-time environment. Moreover, in order to design a control law taking care of the requests of new sources, we use an additional input variable d, accounting for unsatisfied demands, the dynamics of which are constructed as follows. Denote by W h the (continuous) time interval between (h 1)T C and T C, namely W h {t R + :(h 1)T C t ht C }. Let n E (h) the number of new users requesting the use of the resource in W h, namely the flows trying to enter the channel, and by n L (h) the number of users that stop using the resource in W h, namely the flows leaving the channel. Clearly =n(h 1) n L (h). Finally, denote by l N (h) the sum of the minimum levels of transmission of the users trying to access the resource in the time-interval W h and let l A (h) the additional needed resource, namely l A (h) =l N (h) (1 ȳ). We assign to the additional input d, the discrete dynamics described, for k h, by d(k +1)=γd(k)+[l A (h)/]δ(k h), (8) where γ (, 1), δ(k h) =1if k = h and δ(k h) =if k h. The solution of equation (8), for k>his d(k) =γ k h 1 [l A (h)/+γd(h)]. (9) Equation (9) implies, in particular, d(k) > for all k>h, hence d can be used to make the quotes assigned to the active y β α Fig. 1. Time history of the channel occupancy in three different cases First case (bold line): α =.488, β =.1. Second case (dashed line): α =.488, β =1. Third case (solid line): α =.198, β =.1. sources decrease. For instance, we may choose the dynamics of the i-th quote, for i =1,...,n, to be described by x i (k +1)=x i (k)+u i (k) σ i (k)βd(k)/s(k), (1) where β is a positive coefficient, { 1, if xi (k) >, σ i (k) = (11), if x i (k) =, S(k) = σ i(k), and { α[ȳ y(k)]/n (k), if xi (k) <r u i (k) = i, (12), if x i (k) =r i, with 1 α (, 1) and N(k) is the cardinality of L(kT C ). Assumption 1: For t ht C no new request of usage arrives and no user stops using the channel, i.e. the number of active users is constant, and equal to, forallt ht C. If Assumption 1 holds, then for k>hthedynamic equation for the total occupancy of the channel is n(k) y(k +1)= (x i (k +1)+l i )= = y(k)+ α (ȳ y(k)) βd(k). (13) N(k) i L(kT C ) By introducing the variable w = y ȳ, Equation (13) can be rewritten as w(k +1)=(1 α)w(k) βd(k). (14) An example of the time-variation of y for different values of α and β is reported in Figure 1. By numerical evaluation of the minimum position and the minimum value for different values 1 See the proof of Theorem 4.1. Page 112

4 of α and β it can be noted that as α decreases the minimum is reached later and the smaller is the value of α the lower is the value of the minimum. On the contrary, the higher is the value of β the lower is the value of the minimum. Therefore, one can first fix the optimal value of α guaranteeing that the minimum is reached after a prescribed time interval; then the value of β can be tuned to fix the value of the minimum. In order to prove the main results of the article, the following additional assumption is needed. Assumption 2: For t = ht C the system is at the equilibrium, namely w(h) =and d(h) =. Theorem 4.1: If Assumptions 1 and 2 hold, if x i evolves according to the dynamics (1), with d(k) described by (9) and u i (k) described by (12), and if l i + l N (h) < 1, then there exist α (, 1), β> and γ> such that: i) y( k)+l N (h) < 1, forsome k >h, ii) y(k) l i,forallk h. Proof. i) In the domain of the Z transforms, Equations (8) and (14) are D(z) = 1 ( zd(h)+ l ) A(h) z γ and zw(h) βd(z) W (z) =, z 1+α respectively. If γ = 1 α, then the Z transform of w(k) corresponding to the initial conditions w(h) =and d(h) = is W (z) = βl A(h) 1 (z 1+α) 2. By computing the inverse Z transform one obtains w(k) = βl A(h) (k h 1)(1 α) k h 2. (15) If α =ᾱ 1 e 1/m for some m N, thenw(k) reaches the minimum for k = k m + h +1 and we have w( k) = βl A (h)m(1 ᾱ) m 1 /. (16) Finally, if β>β min m(1 ᾱ) m 1, we obtain w( k) < l A (h), what, in turn, implies y( k) =w( k)+ȳ< l A (h)+ȳ = l N (h)+1, hence the claim. ii) The condition l i + l N (h) < 1 implies and, in turn, ȳ l i > ȳ + l N (h) 1=l A (h) l i β max ȳ l A (h) β min >β min. Hence, if β>β min, then (16) yields w(k) > l i ȳ, for all k>h, hence the claim. A result analogous to Theorem 4.1 holds when the control law (12), which is the same for all the fluxes, is substituted with a control law depending on the equilibrium value of the flux to which it is assigned, as the following result shows. Theorem 4.2: The claims of Theorem 4.1 hold, for the same hypotheses, also when (12) is substituted with { α( xi x u i (k) = i (k)), if x i (k) <r i,, if x i (k) =r i. Proof. The dynamic equation for the total occupancy of the channel is y(k +1)= (x i (k +1)+l i )= = y(k)+ α( x i x i (k)) βd(k) = i L(k) = y(k)+α ( x i x i (k)) βd(k) = = y(k)+α(ȳ y(k)) βd(k), which is equal to (14). Therefore the proof of Theorem 4.1 can be repeated. Remark 4.1: The previous result, namely Theorems 4.1 and 4.2 have been obtained by supposing that Assumptions 1 and 2 hold. However, even in a more general case, where several new requests and droppings out occur (in different instants) the method is effective. In fact, if a new request occurs when fluxes are in the decreasing phase an additional impulse appears in (8), what speed up the decreasing process and make new requests more likely to be accepted; if a drop out occurs, then there is more bandwidth available and again new requests are more likely to be accepted. Obviously, several new requests and droppings out affect the precision of equation (16) which is based on the values of l a and n at the instant h in which the first request occur (see again equation (8)). V. CONTROLLER ALGORITHM The previous results, and in particular Theorem 4.1, lead to the following algorithm. STEP.A Choose m N and let α =1 e 1/m.Letγ =1 α. Compute β min and β max and choose β (β min,β max ). Let n be the number of active fluxes at the initial instant. Page 113

5 STEP.B Initialize all the x i,fori =1,...,n, to zero. Initialize d, u i,fori = i...,n,andl A to zero. STEP 1 Evaluate n L, namely the number of sources that stopped using the resource in the interval W h (this quantity is an exogenous input, namely a variable passed to the algorithm from outside it). Evaluate the boolean variable MIS RES (missing resource), which is one if some unsatisfied request occurred in W h (this quantity is also an exogenous input). IF n L OR MIS RES,THEN Update the information table, namely the number of active sources n and the quantities l i and r i for i = 1,...,n. Update F SS, ȳ, l N, l A,and x i,fori =1,...,n. STEP 2 Let x i,fori =1,...,n, evolve according to (1), with u i given by (12), and let d evolve according to (8), namely 2 n n n L, d γ d + MIS RES l A /n, IF x i = r i THEN u i =, ELSE u i = α (ȳ y)/n, x i x i + u i β d, F DA = x 1 /(r 1 l 1 ). Go back to STEP 1. A. Request handling algorithm The overall algorithm to handle new requests can be modeled as follows. If at some time-instant τ a new sender, with minimum level of transmission rate l n+1 and requested transmission rate r n+1, requests to use the channel, DO IF 1 ȳ F DA (r n+1 l n+1 )+l n+1 ACCEPT, Update the information table. Update F SS, ȳ, l N, l A,and x i. x n+1 (τ) =F DA (r n+1 l n+1 ), ELSEIF 1 ȳ l n+1 AND 1 ȳ F (r n+1 l n+1 )+l n+1, DELAY, MIS RES 1, ELSE, REFUSE, MIS RES 1, VI. A SIMULATED SCENARIO To test the effectiveness of the proposed algorithm, a scenario with ten flows described by the parameters specified in Table I has been simulated. The total (not normalized) capacity of the channel is 37. The main events can be described as follows. At the instants 1, 12 and 18 the fluxes F1, F2 and F3 request to use the channel; the request is ACCEPTED and the fluxes are assigned their requested quote, since the sum does not exceeds 37, namely the capacity of the channel. 2 The choice of computing F DA from x 1, l 1 and r 1 in the last assignment is arbitrary; any i {1,...,n} could be used. Flux requests at l i r i drops at F F F F F F F F F F TABLE I PARAMETERS OF THE DATA FLUXES. At time 3 the F4 requests to use the channel; the request is ACCEPTED and F4 is assigned a bandwidth of 7, that is the maximum available. However, since the FDF is updated (see Figure 4), from this instant on F1, F2 and F3 are requested to decrease their bandwidth to leave more band for F4, what is a consequence of the fair distribution policy (see Figure 2 where the instant 3 is denoted by the vertical dashed line on the left). d is kept to zero because there is no missing resource (see Figure 3). At time 5, F5 requests to use the channel; since R 1 (5) + R 2 (5) + R 3 (5) + R 4 (5) = 37 there is no more band to assign and the request cannot be accepted at this time. However, since l 1 + l 2 + l 3 + l 4 + l 5 < 37, the request could possibly be satisfied and is DELAYED. Meanwhile, F1, F2, F3 and F4 are requested to decrease their value (see the vertical dashed line in Figure 2) with the help of the control variable d that is now different from zero (see again Figure 3). In the simulation the value of β is very large and hence the time-behaviour of the fluxes experiences a steep decrement. At time 65, F6 requests to use the channel 3.SinceF1,F2, F3 and F4 are assigned their minimum, there is enough band for F6, which is ACCEPTED and assigned its minimum as well (fairness). At time 7, F7 requests to use the channel. The request is ACCEPTED and F7 is assigned its minimum. From this time on, the control algorithm allows the sources to increase the fluxes up to the full occupancy of the channel at time 15. At time 2 and 21, F1 and F2 drops out, thus making more bandwidth available for the remaining users which are allowed to increase their data fluxes. At time 23, F8 makes a request. All the fluxes are assigned (almost) the band they request since F DA 1 (see Figure 4) and the request is ACCEPTED assigning to F8 (almost) its maximum. At time 24, F9 makes a request, which is DELAYED. At time 25, F1 makes a request, which is ACCEPTED and F1 assigned (almost) its minimum since F DA. Finally, from 25 on, the sources are allowed to gradually increase their fluxes up to the full occupancy of the channel. 3 There is no pre-booking, hence flux F6 is not necessary coming from the same source requesting F5. Page 114

6 d y F1, F F2, F8 F Fig. 3. Time history of d (magnified 5 times) and y. F F F SS F DA F Fig. 4. Time history of F SS and F DA. Fig. 2. Time history of the fluxes. VII. CONCLUSIONS We have described a method to optimize the usage of a shared resource in a dynamic scenario, namely when, asynchronously, new requests occur or current users stop using the resource. The algorithm has been designed in particular for the specific case of data flows in a channel. Its input variables are the minimum and the desired amount of bandwidth requested by each user. The algorithm guarantees both the optimization of the resource and the fair distribution among the users. REFERENCES [1] J.W. Roberts and L. Massoulié, Bandwidth sharing and admission control for elastic traffic, in Proc. of ITC Specialists Seminar, Yokohama, Japan, [2] N. Benameur, S. Ben Fredi, S. Oueslati-Boulahia, and J.W. Roberts, Quality of service and flow level admission control in the internet, Computer Networks, vol. 4, pp , 22. [3] S.Y. Nam, S. Kim, and D.K. Sung, Measurement-based admission control at edge routers, IEEE/ACM Transactions on Networking, vol. 16, no. 2, pp , April 28. [4] S. Jamin, P.B. Danzig, S.J. Shenker, and L. Zhang, A measurementbased admission control algorithm for integrated services packet networks, IEEE/ACM Transactions on Networking, vol. 5, no. 1, pp. 56 7, February [5] Z. Dziong, M. Juda, and L. Mason, A framework for bandwidth management in atm networks aggregate equivalent bandwidth estimation approach, IEEE/ACM Transactions on Networking, vol. 5, no. 1, pp , February [6] F.P. Kelly, P.B. Key, and S. Zachary, Distributed admission control, IEEE Journal on Selected Areas in Communications, vol. 18, no. 12, pp , December 2. [7] P.L. Montessoro and D. De Caneva, REBOOK: a deterministic, robust and scalable resource booking algorithm, Journal of Network and Systems Management, vol. 18, no. 4, pp , 21. Page 115