An Autonomic Framework for Reliable Multicast: A Game Theoretical Approach Based on Social Psychology MARKOS P. ANASTASOPOULOS National Technical University of Athens ATHANASIOS V. VASILAKOS University of Western Macedonia and PANAYOTIS G. COTTIS National Technical University of Athens
A major challenge in wireless terrestrial networks is to provide large-scale reliable multicast and broadcast services. The main problem limiting the scalability of such networks is feedback implosion, a problem arising when a large number of users transmit their feedback messages through the network, occupying a significant portion of system resources. Inspired by social psychology, specifically from the bystander effect phenomenon, an autonomic framework for large-scale reliable multicast services is presented. The self-configuring and selfoptimizing procedures of the proposed autonomic scheme are modeled using game theory. Through appropriate modeling and simulations of the proposed scheme carried out to evaluate its performance, it is found that the new approach suppresses feedback messages very effectively, while at the same time, it does not degrade the timely data transfer. Categories and Subject Descriptors: C.4 [Computer Systems Organization]: Performance of Systems—Reliability, availability, and serviceability; C.2.1 [Computer-Communication Networks]: Network Architecture and Design—Wireless communication General Terms: Algorithms, Reliability Additional Key Words and Phrases: Autonomic communication, autonomic manager, bystander effect, feedback suppression, game theory, Nash equilibrium, reliable multicast, WiMax networks
Author’s addresses: M. P. Anastasopoulos, School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytexneiou Str., 15780, Greece; email:
[email protected]; A. V. Vasilakos, Department of Computer and Telecommunications Engineering, University of Western Macedonia, Kozani 50100, Greece; P. G. Cottis, School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytexneiou Str., 15780, Greece. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or
[email protected]. C 2009 ACM 1556-4665/2009/11-ART21 $10.00 DOI 10.1145/1636665.1636667 http://doi.acm.org/10.1145/1636665.1636667 ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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ACM Reference Format: Anastasopoulos, M. P., Vasilakos, A. V., and Cottis, P. G. 2009. An autonomic framework for reliable multicast: A game theoretical approach based on social psychology. ACM Trans. Autonom. Adapt. Syst. 4, 4, Article 21 (November 2009), 23 pages. DOI = 10.1145/1636665.1636667 http://doi.acm.org/10.1145/1636665.1636667
1. INTRODUCTION As the demand for high-speed ubiquitous Internet access is increasing at a rapid pace, distributed computing systems able to provide Quality of Service (QoS) are gaining increasing popularity. Currently, large-scale computer networks for communications and computation are employed by large companies and institutions to run distributed applications that deal with customer support, Internet control processes, Web content presentation, file sharing, etc. On the other hand, mobile computing is pervading these networks at increasing speed. For example, continually more PDAs and mobile phones are used by employees for Web browsing, video calls, and mail services, causing a tremendous complexity. Manual configuration or control by human operators seems impossible; hence, the current trend is to allow computing systems to manage themselves without direct human intervention. Self-managing includes self-optimization—which is the ability of an autonomic system to optimize its performance—, self-healing, and self-protecting procedures related to the abilities of repairing faulty behavior and detecting intrusions, respectively, and finally, self-configuration. These systems are referred to as autonomic [Dobson et al. 2006; Jacob et al. 2004]. The term was inspired by the autonomic nervous system of the human body that controls important bodily functions, such as respiration, heart beat rate, and blood pressure without any conscious intervention. In large-scale reliable multicast services, one of the main problems to solve is that of feedback implosion [Anastasopoulos and Cottis; Anastasopoulos et al. ]. This problem arises whenever a large number of Subscriber Terminals (STs) transmit FeedBack Messages (FBMs) via the uplink channel. These messages increase linearly with the number of STs and may lead to network congestion. Feedback implosion is a problem well studied in the literature [Obraczka 1998]. In general, solutions to this problem are classified either as structure based or as timer based [Grossglauser 1996]. Structure-based approaches rely on a designated site to process and filter feedback information [Paul et al. 1997; Whetten and Taskale 2000; Holbrook et al. 1995; Baysan and Sarac 2006]. They structure the members of the multicast groups aiming at filtering the amount of feedback. Timer-based solutions rely on probabilistic feedback suppression to avoid implosion at the source [Nonnenmacher and Biersack 1999; Floyd et al. 1997; Crowcroft and Paliwoda 1988; Shuju Wu et al. 2006; Papadopoulos and Parulkar 1995; Anastasopoulos and Cottis]. Receivers delay their retransmission requests for a random interval which may be uniformly, exponentially, or beta distributed between the current time and the Round-Trip Time (RTT) to the source. The aim is that group members located closer to the source send their FBMs sooner, suppressing feedback from farther away members. A site uses periodic session messages to measure its ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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distance to the other group members based on the resulting RTT. An alternative classification of reliable multicast protocols according to the recovery mechanisms used is presented in Ouyang et al. [2005]. The relevant protocols are Automatic Retransmission Request (ARQ)-based, gossip-based, and Forward Error Correction (FEC)-based. In ARQ-based protocols, lost packets are retransmitted until information is received correctly by all receivers. In gossipbased protocols, multicast packets are repeatedly transmitted for a few times by a few of the multicast members in a peer-based mode [Luo et al. 2003]. Finally in FEC-based protocols, redundancy is incorporated in each packet before transmission. The problem of feedback implosion can be treated successfully if a limited number of users send FBMs on behalf of all multicast receivers. Therefore, the feedback suppression problem should be so modeled that, as the number of users increases, their incentive to send FBMs is reduced. This type of modeling feedback suppression is closely related to social psychology, specifically to the bystander effect phenomenon which is diversified in the bystander intervention and the bystander apathy phenomena [Latane and Darley 1969]. With regard to the bystander intervention phenomenon, psychologists have found out that an individual will normally intervene if another individual is in need of help. However, help is less likely to be given by a group of people, a phenomenon known as bystander apathy. The bystander effect is a social phenomenon where people are less likely to intervene in an emergency situation when others are present than when they are alone. The appropriate interpretation of the bystander intervention and bystander apathy phenomena may lead to a simple and effective formulation of the feedback suppression problem. When a small number of users participate in a multicast service, they do not have to suppress their FBMs. The impact of feedback on network performance is insignificant, since a few FBMs do not require significant network resources. On the contrary, if the number of users is high, it is preferable that they avoid sending FBMs rather than being conscientious to send. Considering that even one FBM may help all multicast users to recover from their losses, the contribution of apathetic users to network performance optimization is much more helpful compared to that of conscientious users who send FBMs instantly. Social psychology researchers have modeled the bystander effect phenomenon applying game theory. A typical relevant example would be a scenario where several people are eye witnesses to a crime. An individual would like someone else to call the police and stop the crime because then x units would be added to his payoff. Unfortunately, no one wants to make the call because this action subtracts y ( y < x) units from his payoff. In this article, a new approach to the feedback suppression problem is presented inspired by the aforesaid example. Game theory is used to formulate an autonomic feedback suppression scheme. This game theoretical approach seems a logical choice due to the self-configuring and self-optimizing procedural attributes of autonomic communication [Dobson et al. 2006]. The rest of the article is organized as follows. In Section 2 a brief presentation of game theory basics is given. In Section 3 the self-managing mechanisms related to the feedback suppression problem are modeled using game theory. Then, in Section 4 ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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the autonomic and adaptive system architecture is presented, while in Section 5 an application of the proposed autonomic framework in WiMax networks is presented as a case study. Finally, conclusions are drawn in Section 6. 2. GAME THEORY BASICS Game theory is a branch of applied mathematics which formulates decision making when many entities are involved. While many applications of game theory are related to economics, game theory may also be applied to many other areas ranging from law enforcement and voting rules to biology, psychology, and, recently, engineering. The analysis of the present article is based on noncooperative games where players act individually without exchanging information. 2.1 Definitions The elements of a game are players, strategies, and payoffs. Players are individuals who make decisions. Based on the information obtained with time, they make decisions that aim at maximizing their profit. An action or move made by player i, i ∈ N, N = {1, . . . , N }, denoted by ai , is a choice he can make. The rule telling which action to choose at the decision instants of the game is called strategy, si . The nonempty set Si denotes the set of all strategies available to player i. Hence, the Cartesian product, S = S1 × S2 × · · · × SN , sometimes called strategy space of the game [Osborne and Rubinstein 1994], is the set of all players’ strategies. The strategy profile s is the vector containing the strategies of all players: s = (si )i∈N = (s1 , s2 , . . . , sN ). For any strategy profile s ∈ S, a payoff ui (s) is associated with player i which constitutes the expected utility that a player receives as a function of the strategies chosen by him and the other players. In economics, the payoffs are usually profits or consumer utilities, while in biology the payoffs usually represent the expected number of surviving offspring. Often in the literature, a game is written in normal form denoted by a triplet G = (N, S, U), where U is a vector containing the payoffs of all players. 2.2 Equilibrium and Dominant Strategies A very important concept in game theory is that of equilibrium. Equilib∗ rium s∗ = {s1∗ , . . . , sN } is a strategy combination comprising the best strategy from each of the N players in the game [Rasmusen 1989]. The equilibrium strategies are the strategies that players select in their effort to maximize their individual payoff. If (s1 , . . . , sN ) is a strategy profile, the notation s−i = (s1 , . . . si−1 , si+1 , . . . , sn ) denotes the combination of strategies of all players except player i. s−i is of utmost importance because it helps player i to choose its best available strategy. The best response or best reply of player i to the strate∗ gies chosen by the other players is the strategy s−i that yields the highest payoff ∗ for him. The strategy s−i is a dominant strategy if it is a player’s strictly best response to any combination of strategies the other players might choose, in the sense that, whichever strategy they choose, his payoff is the highest under ∗ the choice of s−i . A player’s inferior strategies are called dominated strategies. ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Finally, dominant strategy equilibrium is a strategy combination consisting of the dominant strategies of all players. Not many games have a dominant strategy equilibrium. Nevertheless, dominance proves useful even when it does not resolve things clearly. In such cases, the concept of iterated dominance equilibrium is used. To this end, weak dominance should be defined. Strategy si is weakly dominated if some other strategy si exists which may be better but never worse, that is, it may yield a higher payoff in some strategy profile but never a lower payoff. An iterated dominance equilibrium is a strategy combination determined by deleting a weakly dominated strategy from the strategy set of one of the players, recalculating to find which remaining strategies are weakly dominated, deleting one of them, and continuing the procedure until one strategy remains for every player [Rasmusen 1989]. However, the majority of games lack even iterated dominance equilibrium. In these cases, Nash Equilibrium (NE) is used. A strategy profile is a NE if it is best reply to itself [Osborne and Rubinstein 1994]. When a game is in NE, no player has the incentive to change his strategy provided that the other players do not. Concluding, every dominant strategy is a NE, but not every NE is a dominant strategy equilibrium. If a strategy is dominant, it is best response to any strategies the other players may choose, including their equilibrium strategies. If a strategy is part of a NE, it needs only to be in response to the equilibrium strategies of the other players. 2.3 Pure and Mixed Strategies So far, the analysis has been limited to finite sets of players’ actions. It is often useful and realistic to expand the strategy space to include random strategies. The latter strategies are called mixed, while the former are called pure strategies. In other words, a mixed strategy of player i is a probability distribution over his set of pure strategies [Rasmusen 1989]. Mixed strategies are infinite, even if strategies sets are finite, due to the fact that the assigned probabilities are infinite. 2.4 Existence of Equilibrium In finite games the following theorem holds. THEOREM 2.1. Any finite strategic game has a Nash equilibrium in either mixed or pure strategies. The proof of Theorem 2.1, also known as the Nash theorem, is based on the Kakutanis fixed point theorem. Further details may found in Gibbons [1992]. 3. MODELING THE SELF-CONFIGURING AND SELF-OPTIMIZING PROCEDURES USING GAME THEORY 3.1 The Two-Player Feedback Suppression Game Consider the problem of feedback suppression as a game belonging to the general class of contribution games [Osborne and Rubinstein 1994; Rasmusen ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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M. P. Anastasopoulos et al. Table I. Feedback Suppression Game
Player 1
No FBM ( p) FBM (1 − p)
No FBM ( p) (0, 0) (1 − dE/α, 1)
Player 2 FBM (1 − p) (1, 1 − dE/α) (1 − dE/α, 1 − dE/α)
1989]. This classification is used to describe games where, though the players have the choice of contributing to the public good, they prefer that another player contributes. In a feedback suppression game, a player who has lost packets would like to send a FBM asking for packet retransmission because by retrieving its lost packets his QoS constraints are satisfied. However, no player wants to send a FBM because this action has a certain cost. Since energy issues are crucial for the survivability of wireless networks, in the proposed game theoretical formulation of the feedback suppression problem the cost of sending a FBM is related to the respective energy consumption and the residual battery capacity of each node. Table I shows the feedback suppression game between two players who have lost the same packet. Let dE denote the energy consumption when a FBM is sent and α the residual battery power at the initialization of the multicast service. In the present approach it is assumed that the relative cost of a player to send a FBM is dE/α while the maximum payoff has been normalized to 1. A player will not send a FBM if he believes that the other player will send. In this case, the first player receives a maximum payoff equal to 1, while the payoff of the other player is reduced by the relative cost of sending a FBM. When none of the players sends a FBM, their payoff is set at 0 to emphasize the importance of sending a FBM, since if no user sends a FBM, the lost packets are not retrieved and the multicast service is terminated. The preceding game has two asymmetric pure strategy equilibria, (1 − dE/α, 1), (1, 1 − dE/α), and a symmetric mixed-strategy equilibrium. 3.2 Extension to N Players The previous approach may be extended to cover the case where N players participate in the game. The basic elements of the feedback suppression game are players, strategies, and payoffs. A node that wishes to send a FBM to the source in order to retrieve its lost packets is a player of the game. Usually, the number of users in feedback suppression problems are estimated through the number of FBMs that arrive at the source [Nonnenmacher and Biersack 1999]. In terrestrial wireless networks operating in the frequency range 10– 66 GHz such as WiMax, the problem appears when the multicast system is under rain conditions. Then, the majority of users located within the served area suffer simultaneously from a high number of packet losses; therefore, the number of players is quite close to the total number of multicast receivers [Anastasopoulos and Cottis]. In this formulation, too, the strategy set of player i , i ∈ N, is Si = {No FBM, FBM}. The payoffs are determined as follows: if no player sends a FBM, the payoff of all players including player i is 0; if player i sends a FBM, his payoff is 1 − d E/α; if at least one of the other N − 1 players ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Fig. 1. Impact of users population on the transmission probability of FBMs.
sends a FBM while player i does not send, his payoff is 1. The N-player game has also an asymmetric pure-strategy and a symmetric mixed-strategy equilibrium. To deal with the N -player version of the multicast game, let p denote the probability that player a does not send a FBM. The probability that no other player except for player i sends a FBM is p N −1 . Therefore, the probability that at least one of the other N-1 players sends a FBM is 1 − p N −1 . Equating the pure strategy payoffs of player i using the payoff-equating method [Osborne and Rubinstein 1994] πplayeri
sends a FBM
= πplayeri
does not send a FBM
the following equation is obtained for the equilibrium probability peq . N −1 N −1 N −1 N −1 (1 − dE/α) · peq + (1 − dE/α) · 1 − peq + 1 · 1 − peq = 0 · peq
(1)
(2)
or peq = (dE/α)1/(N −1)
(3)
The probability that at least one user sends a FBM depends on the relative cost dE/α and is given from PFB (dE/α) = 1 − (dE/α)1/(N −1) .
(4)
It is clear from (4) that, if the number of users is low, PFB is high. In this case, multicast receivers react consciously according to the bystander intervention phenomenon and send a FBM. However, as N increases, PFB is reduced and users gradually lose their conscious behavior. Hence, feedback suppression is increased. The aforesaid trend is clearly depicted in Figure 1, where the probability that a user sends a FBM is plotted versus the number of users. ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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3.3 The Genovese Syndrome: The Need for Backup Mechanisms Though feedback suppression may be modeled solely on the bystander effect, when the number of multicast receivers is high, the nodes become so indifferent that no FBMs are sent. This is clearly demonstrated by the plot of Figure 1. A similar phenomenon of social indifference is described by the Genovese syndrome or Genovese effect when, in the presence of many people, no one intervenes to prevent a criminal action [Latane and Darley 1969]. In the feedback suppression problem, the Genovese syndrome describes the situation when no FBM is sent. The respective appearance probability is given by N PGEN (dE/α) = 1 − PFB (dE/α) = (dE/α) N /(N −1) . (5) From (5) it is easily verified that when the relative cost is negligible (dE/α → 0), the appearance probability of the Genovese syndrome is zero. However, when the cost becomes high (dE/α → 1), it is very likely that the Genovese syndrome will appear. Taking into consideration the previous arguments, it is evident that the exploitation of the bystander effect alone cannot ensure reliable data transfer, since, if the Genovese syndrome appears, no player will send a FBM. To deal with the possible appearance of the Genovese syndrome, appropriate backup mechanisms should be provided. A simple backup mechanism is based on the use of timers. Each multicast receiver employs a timer configured to expire after RTT, that is, the delay from the transmission of the FBM until the reception of the retransmitted packets. This delay is the sum of the propagation delay plus the packet processing time. Assuming that all STs are identical and given that propagation delays in wireless terrestrial networks are negligible compared to the packet processing time, the RTTs of all STs are equal to each other. If the timer expires before the recovery of the lost packets, the terminals send a FBM asking for packet retransmission. The timers should be appropriately adjusted to provide sufficient time for the retransmitted packets to reach all multicast receivers so that they suppress their FBMs. Taking into account both the bystander effect and the Genovese syndrome, the expected number of FBMs is determined by EFB (dE/α) = PFB (dE/α) · N + PGEN (dE/α) · N = 1 − (dE/α)1/(N −1) + (dE/α) N /(N −1) · N .
(6)
As shown in the Appendix, EFB (dE/α) is minimized at (dE/α)∗ = 1/N . Due to the delay introduced by the timers employed when the relative cost of sending a FBM is high, to deal with the possible appearance of the Genovese syndrome, the average feedback delay is determined by DFB (dE/α) = PFB (dE/α) · 0 + PGEN (dE/α) · RTT = (dE/α) N /(N −1) · RTT.
(7)
As easily deduced from (6) and (7), if dE/α → 1, EFB → N , leading to feedback implosion and DFB → RT T which is the maximum delay. To prevent ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Fig. 2. The game theory feedback suppression scheme: At t = t0 the feedback procedure is initiated. If no FBM has been received up to t = t0 + RTT, the backup mechanism is activated.
feedback implosion from happening, an appropriate combination of the feedback suppression method just analyzed with the timer-based feedback suppression schemes presented in Obraczka [1998] and Nonnenmacher and Biersack [1999] is proposed. In this course, instead of delaying the transmission of FBMs for a time period equal to RTT, the following scheme is proposed. (i) The users wait for RTT and then schedule exponentially distributed random timers characterized by parameters ti , i ∈ N, randomly selected in the interval [0, T ]. The exponential distribution of the timers’ duration is defined from Nonnenmacher and Biersack [1999] λ 1 · e T ti 0 ≤ ti ≤ T λ f ti (ti ) = e −1 (8) 0 otherwise, where T is a maximum delaying period. (ii) When its timer expires, a multicast receiver proceeds to the following actions: (a) If the retransmitted packets have not been received up to that instant, it sends a FBM requesting for retransmission. (b) If the retransmitted packets have already been received, it does not send a FBM. The previous procedure is depicted in Figure 2. At t = t0 , a feedback cycle begins and every user decides whether to send a FBM or not. If the retransmitted packets have not been received up to t = t0 + RT T , the backup mechanism is activated and probabilistic feedback suppression is performed. In this case, the expected number of FBMs is given from (exp)
EFB (dE/α, T ) = PFB (dE/α) · N + PGEN (dE/α) · EFB ,
(9)
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where (exp)
EFB
⎛ ⎞ N RT T T −λ RT T 1 − e eλ T −1 RT T = N λ−1 − eλ T ⎝ − 1⎠ 1 − e−λ e
(10)
is the expected number of FBMs when exponential backup timers are employed. Moreover, the average feedback delay when the proposed Game Theory Feedback Suppression (GTFS) scheme is combined with the timer-based suppression scheme is estimated from (exp) DFB (dE/α, T ) = PFB (dE/α) · 0 + PGEN (dE/α) · RTT + DFB , (11) where (exp)
DFB
T
1−
= 0
eλm − 1 eλ − 1 (exp)
N dm.
(12)
(exp)
Details concerning the evaluation of EFB and DFB may be found in Nonnenmacher and Biersack [1999]. Taking into account (5) and (12), (11) yields N T N /(N −1) eλm − 1 DFB (dE/α, T ) = dE/α RTT + 1− λ dm . (13) e −1 0 In Figure 3, EFB (dE/α) is plotted for various receiver populations applying the GTFS scheme alone. The relevant curves are convex having a minimum at (dE/α)∗ = 1/N (see the Appendix). When dE/α < (dE/α)∗ and as long as dE/α is away from (dE/α)∗ , the expected number of FBMs is rather high since the relative cost of sending a FBM is negligible. In this case, users have a higher benefit by sending a FBM rather than by expecting another user to do it on their behalf. In other words, why should receivers suppress their FBMs and probably risk their QoS, when their relative cost is insignificant? Therefore, some receivers will immediately send a FBM suppressing the feedback from the others. This situation confirms the analogy with the social phenomenon of bystander effect. Also, for low dE/α values, the three curves are very close to each other because the cost of sending a FBM is low and the backup mechanisms remain idle. In the theoretical case dE/α → 0, EFB → N ; then all receivers will instantly send a FBM since the relative cost is zero. On the other hand, when dE/α > (dE/α)∗ the cost of sending a FBM becomes significant. In this case, the users are reluctant to send FBMs and prefer to risk that some other users will; then, the Genovese syndrome appears. As dE/α moves away from (dE/α)∗ , the receivers tend to send FBMs since the risk that the multicast service will be terminated becomes high. Theoretically, as dE/α → 1 all receivers will send a FBM. Results concerning the application of the GTFS scheme combined with exponential timers are plotted in Figures 4 and 5. In Figure 4, EFB (dE/α) is plotted for various values of T . In this case, the three curves are minimized (exp) at (dE/α)∗ = 1/EFB (see the Appendix). It should be noted that due to the synergy of the timer operation the maximum number of FBMs, occurring when ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Fig. 3. Expected number of FBMs of the GTFS scheme versus dE/α for various values of receiver population: (a) dE/α ∈ [10−100 , 100 ]; (b) dE/α ∈ [10−10 , 100 ].
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Fig. 4. Expected number of FBMs versus dE/α for different values of the delaying period T (N = 1000, λ = 10).
Fig. 5. Average normalized delay versus dE/α (N = 1000, λ = 10). ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Fig. 6. Expected number of FBMs of the GTFS scheme and of various exponential timer-based schemes versus the receiver population (λ = 10).
the cost dE/α → 1, is significantly reduced compared to N . The trade-off for this reduction is that the average feedback delay increases as shown in Figure 5, where the average delay normalized with respect to RTT is plotted as a function of dE/α. For low dE/α values, the delay is nearly zero, since the probability of a feedback transmission is nearly 1. On the other hand, for large values of dE/α → 1, the average delay is increased since the possibility appearance of the Genovese syndrome activates the backup mechanisms. In Figure 6, the proposed GTFS scheme is compared to the exponential timerbased feedback suppression scheme presented in Nonnenmacher and Biersack [1999]. The GTFS scheme achieves a considerably better suppression performance compared to schemes based solely on exponential timers. Only timerbased schemes having a large delaying period (and, consequently, a high average delay) perform comparably to the proposed GTFS scheme. In Figure 7, the average delay of feedback transmission normalized with respect to RTT is plotted versus the number of receivers. The superiority of the proposed GTFS scheme compared to exponential timer-based schemes is evident with regard either to the expected number of FBMs or to the average delay. Finally, in Figure 8 the normalized delaying period T/RTT necessary to keep the number of FBMs below a certain threshold Eth is plotted versus dE/α. It is observed that, as the relative cost dE/α increases, the delaying period and, consequently, the average timer duration should be prolonged. It is also observed ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Fig. 7. Average normalized delay versus the number of receivers.
Fig. 8. Normalized delaying period necessary to keep the number of FBMs below Eth plotted as a function of dE/α for the hybrid GTFS timer-based scheme. ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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that as the number of users increases, the backup mechanism is activated at lower relative cost levels. However, after the activation of the backup mechanism, the estimated delaying period does not vary significantly with the number of users. This observation is important as it reveals the tolerance of the proposed scheme with regard to a inaccurate estimation of the receiver’s population. 4. THE AUTONOMIC AND ADAPTIVE SYSTEM ARCHITECTURE In this section an autonomic and adaptive framework for reliable multicast services is presented along with a hybrid feedback suppression scheme for an autonomic and adaptive system architecture. 4.1 Autonomic Communication Systems Autonomic communication systems manage themselves to automatically carry out tasks otherwise performed through human intervention [Litoiu 2007]. In self-managing systems, an automated method exists to retrieve all necessary information, analyze the collected data, plan a sequence of actions specifying the necessary changes, and, finally, perform these actions [IBM 2006]. In a typical intelligent autonomic control there are sensors to collect information from the environment and effectors to change the configuration of an element. The combination of sensors and effectors forms the managing interface. Furthermore, the autonomic architecture is composed of four procedures that share knowledge: monitoring, which collects, aggregates, and filters information; analyzing, which provides functions that allow the autonomic manager to learn about the IT environment and predict future situations; planning, which plans the actions required to achieve the objectives; and executing, which controls the execution of a plan. 4.2 The Self-Managing Procedure In this section an autonomic communications framework for large-scale reliable multicast services is presented. Its objectives are, first, to limit the number of FBMs below a certain threshold Eth and, second, to prolong system survivability. To accomplish these objectives, the autonomic manager should perform the following actions. (1) Monitoring. The multicast source informs multicast receivers about the number N of the users served and the number of FBMs. It is assumed that every receiver is also aware of its battery capacity, α, and of the energy cost to send a FBM, dE. (2) Analyzing, Planning and Executing. Based on the delay that FBMs arrive at the source, the following two cases are considered. Case I. FBMs arrive at the source with an RTT delay. In the beginning of the multicast service when the majority of users have abundant power, the average relative cost, dE/α AV , is very low and the delaying period of the backup mechanisms is set at zero (T = 0). In this case, FBMs arrive at the source simultaneously with an RTT delay. If the number of users is high, feedback implosion may appear, since the cost ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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for sending a FBM is negligible. If the number of received FBMs is less than Eth , the GTFS scheme is followed and PFB is given by (4). Otherwise, EFB is set at Eth . Employing the approximation (1 − x) N ≈ 1 − N x when x 1, (6) yields Eth = PFB · N + (1 − N · PFB ) · N .
(14)
Thus, PFB is given by PFB =
N − Eth . N (N − 1)
(15)
Concluding, when the FBMs arrive at the multicast source with an RTT delay, probabilistic feedback transmission is done with FBM probability given either by (15), when the number of FBMs received by the multicast source exceeds Eth , or by (6), when this number is less than Eth , following the proposed GTFS scheme. The transition from one transmission probability to the other is imposed by the source. Case II. FBMs arrive at the source with at least 2RTT delay. As dE/α AV increases, the Genovese syndrome appears and the number of FBMs starts to increase. If the backup timers were not activated, the number of FBMs would be left to take its limiting value equal to N . To keep the number of FBMs below Eth , a real-time adaptive algorithm appropriately adjusting the maximum delaying period of the timers, T , is given shortly. Note that F BM (k) is the number of FBMs measured at the multicast source and T (k) , (dE/α)(k) AV are the maximum delaying period of the backup timers and the average relative cost evaluated by the multicast source at the k th cycle of the feedback scheme. (1) Initialization: Set the maximum number of FBMs at Eth . Set T (0) at 0. (2) k th cycle of the multicast protocol: (a) if FBM(k) < Eth set T (k) at 0 and determine (dE/α)(k) AV employing (6), that is, by solving (k) 1/(N −1) (k) N /(N −1) FBM(k) = 1 − dE/α AV · N. + dE/α AV (16) (b) if FBM(k) ≥ Eth (i) determine T (k) employing (9), that is, by solving
Eth = ·
1/(N −1)
N /(N −1) 1 − (dE/α)AV · N + (dE/α)(k−1) AV ⎧ ⎞⎫ ⎛ N T −1 −λ RT(k)T ⎨ eλ RT ⎬ T (k) T 1 − e λ RT(k)T −e T ⎝ − 1⎠ . (17) λ−1 −λ ⎩ e ⎭ 1−e (k−1)
Under this value T (k) , the backup mechanism will appropriately delay the feedback transmission so that the average number of FBMs will not exceed Eth . ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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(ii) estimate (dE/α)(k) AV employing (9), that is, by solving
FBM(k) = ·
1/(N −1) N /(N −1) 1 − (dE/α)(k) · N + (dE/α)(k) AV AV ⎧ ⎞⎫ ⎛ N T −1 −λ RT(k)T ⎨ eλ RT ⎬ RT T T (k) T 1 − e λ (k) ⎠ . ⎝ T − e − 1 (18) ⎩ eλ−1 ⎭ 1 − e−λ
It is also worth mentioning that the proposed autonomic scheme behaves as the central nervous system. As the human system increases the heart beat rate when someone runs, the proposed scheme increases the maximum delaying period of the timers to keep the average number of FBMs below a certain threshold. Thus, as the average power decreases—as indicated by the respective increase of the average relative cost of sending a FBM, (dE/α)AV −, the average lifetime of the autonomic communications system is prolonged. This is achieved at the cost of delaying the overall multicast procedure.
5. CASE STUDY: APPLICATION IN WIMAX NETWORKS The proposed autonomic framework has been applied to a typical terrestrial wireless network such as WiMax. The network under consideration is assumed to operate at 40 GHz, where the dominant factor impairing link performance is rain attenuation. For this reason, a dynamical rain rate field has been implemented. The rain medium exhibits both spatial and temporal variations. For the simulation, the spatial variations of rain were simulated using HYCELL, a model for rain fields and rain cells structure developed by ONERA [F´eral et al. 2003a, 2003b]. HYCELL is used to produce two-dimensional rain rate fields, R(x, y) over an area of about 100 × 100 km2 ; that is R(x, y) the rainfall rate at a point (x, y). These rain fields follow the local climatic conditions. Moreover, the temporal variations of the wireless channel, R(t), were simulated using the methodology described in Panagopoulos and Kanellopoulos [2003], where R(t) denotes the rainfall rate at time t. Having implemented the appropriate model for R(x, y; t), the next step is to determine Aij (t), which is the rain attenuation induced along link (i, j ) at time t. This is achieved by integrating the specific rain attenuation A0 (x, y; t) (in dB/km) over the link path connecting i to j taking into account the local properties of the rainfall medium through R(x, y; t). L ij Aij (t) = A0 (x, y; t)dl (19) 0
where A0 (x, y; t) = a R(x, y; t)b
(20)
The parameters a, b employed in (18) are related to the frequency, elevation angle, incident polarization, temperature, and raindrop size distribution [ITUR P 2003]. ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Fig. 9. Average number of FBMs versus (dE/α)AV for Eth = 10(λ = 10).
To keep the analysis tractable, binary PSK modulation is assumed. Therefore, the BER of link (i, j ) is BERij = Q (21) 2 · 10(Eb /N0 ) j /10 , where
1 z erfc √ (22) 2 2 and (Eb/N0 ) j is the bit energy-to-noise spectral density ratio at the decoder input of node j under clear sky conditions. This ratio depends on the transmitter power, the receiver antenna gain, and the free space loss. Under rain conditions, Aij [dB] must be taken into account in the evaluation BE Rij . Then, (19) yields 2 · 10[(Eb /N0 ) j −Ai j ]/10 . (23) BERij = Q Q(z) =
A similar simulation model has been also applied in Anastasopoulos et al. [2008] and verified using Slavin et al. [2007]. Due to the adverse propagation conditions when it rains, packets are lost, necessitating the transmission of FBMs. Then, the autonomic feedback suppression mechanism is activated. In Figures 9 and 10, the average number of FBMs and the corresponding normalized maximum delaying period are plotted versus (dE/α)AV as given when Eth = 10. With regard to Figures 9 and 10, two critical values of (dE/α)AV should be mentioned which are evaluated by equating EFB (dE/α) from (6) with ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Fig. 10. Average delaying period versus (dE/α)AV for Eth = 10.
the threshold value Eth , that is, by solving ! 1/(N −1) N /(N −1) Eth = 1 − (dE/α)AV · N. + (dE/α)AV
(24)
From these two (dE/α)AV values, the higher one, denoted by (dE/α)ACT , determines the initialization of the delaying procedure performed through backup timers, while the lower one, denoted by (dE/α)PR , determines the (dE/α)AV range where FBMs arrive at the source with an RTT delay and their number may exceed Eth . Taking these values into consideration, the behavior of the autonomic framework with regard to the average relative cost of sending a FBM is determined as follows. (i) (dE/α)AV < (dE/α)PR . Probalistic transmission of FBMs is imposed by the autonomic framework according to (15) forcing EFB to remain equal to the threshold level Eth . (ii) (dE/α)PR < (dE/α)AV < (dE/α)ACT . Feedback suppression is done according to the GTFS scheme. (iii) (dE/α)ACT < (dE/α)AV . The backup timers are activated by the autonomic framework to appropriately delay the feedback procedure so that EFB is restricted at Eth . For the case study examined, the critical values of (dE/α)AV are given in Table II. From Figure 9 it is readily observed that for (dE/α)AV < (dE/α)PR , the the average number of FBMs is kept equal to Eth due to the self-optimizating behavior ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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M. P. Anastasopoulos et al. Table II. Critical Values of (dE/α)∗AV for the Case Study Examined N = 102 N = 103
(dE/α) P R 3 · 10−5 4 · 10−5
(dE/α) AV 10−2 10−3
(dE/α) ACT 7 · 10−2 4.5 · 10−3
Fig. 11. Impact of inaccurate estimation of receivers population for Case I. (a) Expected number of FBMs versus receivers population for Eth = 10 and Eth = 15 when N is estimated with a 10% error. (b) Expected number of FBMs with respect to the estimation error for N = 2000.
of the multicast network achieved through appropriate probabilistic transmission of FBMs performed by every receiver. However, when the average baterry power is low (that is, the relative cost of sending a FBM exceeds (dE/α)ACT ), the appearance probability of the Genovese syndrome is high. Hence, to restrict the number of FBMs below the threshold level Eth , the feedback procedure is delayed at every receiver by its backup mechanism through appropriate adjustment of the maximum delaying period. From Figure 10 it is observed that, as the number of users increases, the backup mechanisms are activated by the source at lower values of (dE/α)AV . One should also note the convergence of simulated and analytical results. ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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Fig. 12. Impact of inaccurate estimation of receivers population for Case II. (a) Expected number of FBMs versus receivers population for Eth = 10 and Eth = 15 when N is estimated with 10% error; (b) expected number of FBMs with respect to the estimation error for N = 2000.
Finally, how a possible erroneous estimation of the number of receivers affects the expected number of FBMs is investigated in Figures 11 and 12. In Figure 11 the performance of the proposed autonomic scheme is examined when FBMs arrive at the source with an RTT delay (Case I) assuming that the number of receivers is erroneously estimated. It is observed that for high values of N the error is reduced since the performance of the proposed scheme is not affected by N when N is large. This is also verified from Figure 3, where for low values of dE/α the expected number of FBMs is almost the same regardless of N when N ≥ 103 . Also, a higher error is observed when the receiver’s population is overestimated since, due to the relevant reduction of PFB , the backup timers are activated more frequently. The tolerance of the proposed scheme is also confirmed by the plots of Figure 12 concerning the case when FBMs arrive at the source with at least double RTT delay (Case II). It is observed that for an 30% error in the estimation of a population of 2000 receivers, the number of FBMs does not exceed 15.5 instead of the threshold level set at Eth = 15. ACM Transactions on Autonomous and Adaptive Systems, Vol. 4, No. 4, Article 21, Publication date: November 2009.
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6. CONCLUSIONS An autonomic framework for large-scale multicast services has been proposed. The self-managing mechanisms of the proposed autonomic system are modeled using game theory. In the beginning of the feedback suppression procedure, based on the number of multicast receivers and their residual battery power, the users unilaterally decide whether to send a feedback message or not. When the number of multicast receivers is high, users behave as apathetic human beings; thus feedback implosion is avoided. If, due to reduction of power supply, the relative cost of sending FBMs renders the receivers reluctant to send FBMs, backup mechanisms are activated to ensure the continuity of the multicast service. A real-time adaptive scheme concerning multicast WiMax networks was presented to limit the number of feedback messages below a certain threshold. APPENDIX The expected number of FBMs, EFB , is minimized at (dE/α)∗ = 1/EFB . Recall that (exp)
EFB (dE/α) = 1 − (dE/α)1/(N −1) · N + (dE/α) N /(N −1) · X ,
(25)
(exp)
where X = N (GTFS alone) or X = EFB (GTFS combined with an exponential timer). After straightforward algebra (21) yields ∂ EFB (dE/α) ∂(dE/α) ! ! ∂ ∂ = 1 − (dE/α)1/N −1 · N + (dE/α) N /(N −1) · X ∂(dE/α) ∂(dE/α) N N 1 1 =− (dE/α) N −1 −1 + (dE/α) N −1 · X N −1 N −1 ! N 1 =− (dE/α) N −1 (dE/α)−1 − X . N −1 Hence, the value (dE/α)∗ = 1/ X makes shown that
∂ 2 (dE/α)∗ ∂ 2 (dE/α)
∂[EFB (dE/α)] ∂(dE/α)
equal to 0. It may easily be
> 0. Hence, (21) has a unique minimum at (dE/α)∗ = 1/ X .
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