Sensitivity Analysis for an Optimal Routing Policy in an Ad Hoc Wireless Network

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1 1 Sensitivity Analysis for an Otimal Routing Policy in an Ad Hoc Wireless Network Tara Javidi and Demosthenis Teneketzis Deartment of Electrical Engineering and Comuter Science University of Michigan Ann Arbor, MI Abstract We examine the sensitivity of otimal routing olicies in ad hoc wireless networks with resect to estimation errors in channel quality. We consider an ad hoc wireless network where the wireless links from each node to its neighbors are modeled by a robability distribution describing the local broadcast nature of wireless transmissions. These robability distributions are estimated in real-time. We investigate the imact of estimation errors on the erformance of a set of roosed routing olicies. I. INTRODUCTION As the size of communication networks increases and the alications of such networks sreads to various fields, resource allocation issues such as connection admission control, routing, etc., become an increasingly imortant comonent of communication research. Ad hoc networks research consists of a large body of work which addresses these issues in the context of networks with no central controller and unsecied connectivity toology, where each node can itself act as a store-and-forward router (see [1]). The general routing roblem in an ad hoc network is to define a olicy which, given the trajectory histories of all the messages, chooses the nodes to transmit the messages next. Such a olicy must be imlemented in a distributed fashion. There has been an extremely rich literature on routing in ad hoc networks (for a summary see [1]), where several centralized and distributed routing algorithms are roosed and notions of shortest ath and minimum energy routing have been established. Various objectives can be considered in a network routing otimization, including caacity, timeliness, and energy consumtion. Further challenges are osed by wireless networks, where similar goals must be achieved with unreliable, time-varying channels, and where new concerns, such as energy consumtion and channel interference imose additional constraints. It is shown (see [2] and [3]) that in wireless networks due to the unreliability of channels retransmission mechanisms have a non-negligible imact on the energy erformance of routing rotocols. In other words, the roosed routing olicies (e.g. MRPC in [3]) achieves a better erformance by including the channel quality and exected number of retransmissions, both functions of robability of successful transmission, in the definition of each link s length or each node s transmission cost. Furthermore, in [4], [5], and [6] it is shown that the local broadcast nature of wireless transmission, which usually causes interference among neighboring nodes, can be used to our advantage. This can be done successful transmissions are acknowledged by the nodes which receive the message. Using the acknowlegements, samle-ath-deendent otimal routes can be constructed (see Section II). The construction of samle ath deendent routing as well as MRPC relies on the estimates of channel quality to achieve an imroved erformance in ad hoc routing. In other words, the routing in wireless systems is best addressed by a stochastic model of the system. In general, service rovisioning and resource allocation issues (such as admission control, routing, etc.) in wireless networks are best modeled as stochastic scheduling and stochastic control roblems, where the wireless links are described by stochastic rocesses. The statistic of any wireless link deends on the hysical channel (additive noise, ath loss, shadowing, fading, etc.), the number of users that use the link simultaneously, and the users transmission strategies. Generally, the overall structure and statistical behavior of the system, e.g. the marginal

2 2 Source c c c c c5 Destination R 56 Fig. 1. Ad hoc wireless network with robabilistic local broadcast model and joint distributions of the rocesses involved, is studied and modeled off-line, while the articular arameters of such models, e.g. mean and covariance, are left to be estimated in a real-time measurement-based fashion. For instance, a single-ho wireless link might be modeled as an indeendent identically distributed binary symmetric channel, whose transmission error robability is estimated on-line. The existence of on-line estimation adds a new dimension to the question of otimal routing. Ideally, routing as well as other resource allocation mechanisms can otentially rovide information on the statistics of the wireless channels since any resource allocation mechanism regulates the communications in the system. In fact, there exists a trade-off between identication and learning via communication versus minimizing the communication costs. Hence, even in situations where the wireless system is controlled in a centralized manner, the estimation roblem combined with the control issues, should be ideally studied as a stochastic control roblem with imerfect information. Stochastic control roblems with imerfect information are dual control roblems that address joint estimation and control issues. The information state [7] for these roblems lies in an infinite dimensional sace even when the state-sace and action sace are finite. This feature makes such dual control roblems analytically and comutationally dficult. An alternative aroach is to decoule the estimation and control issues in dual control roblems. Such an aroach rovides a arameter estimation algorithm which oerates indeendently of the control decisions and feeds the estimated arameters into a controller designed under the erfect information assumtion. Following such an aroach, wireless ad hoc routing can be addressed by the following three ste rocedure: ( ) Off-line modeling of the overall statistical behavior of the wireless links; ( ) secication of the channel quality via real-time and measurement-based estimation of robabilities of successful transmissions at all nodes;; and ( ) determination of otimal decision (e.g. routing) strategies assuming that the results of stes ( ) and ( ) describe the true stochastic behavior of the broadcast channels (referred to as broadcast model). The construction of otimal samle ath deendent routing strategies as well as MRPC (see [2], [4], [5], [6]) are examles of the solution to ste ( ) of the wireless ad hoc routing roblem. In the aroach descrbied above, there are errors associated with the estimation techniques used in ( ), and the accuracy of the estimated arameters is limited to the emloyed estimation algorithm s error margin. Furthermore, the otimal decision (e.g. routing) strategy resulting from ( ) is guaranteed to be otimal only for the articular values of arameters given by ( ); and it generally varies when these arameters change. Hence, it is vital to erform sensitivity analysis in order to quanty the loss in erformance of the roosed decision strategy in the resence of the aforementioned estimation errors. This fact imlies that a major comonent of design and analysis in any wireless system is the sensitivity analysis with resect to errors in channel modeling and channel estimation. In this aer we resent a sensitivity analysis of a known otimal (with resect to an energy consumtion

3 3 criterion) routing olicy in a stochastic ad hoc network. Our analysis is based on the model and results of [4],[5], and [6]. In these aers, the authors investigate a network routing roblem where a robabilistic model for wireless local broadcasts is used. Under the assumtion that the transmission robabilities of the local broadcast model for each node are time-invariant and recisely known, the existence of an otimal riority olicy with time-invariant indices is shown in [4], [5], and [6]. As exected, these indices deend on the arameters of the local broadcast model. We investigate the sensitivity of this riority olicy with resect to errors in the knowledge of the aforementioned transmission robabilities, and analytically determine the imact of errors in the broadcast model on the erformance of the otimal olicy. We quanty this imact as follows: ( ) we first establish aroriate distance measures between two robabilistic broadcast models, and between two olicies in terms of their erformance; ( ) we construct olicies and that are otimal for the true broadcast model and the estimated broadcast model, resectively; and ( ) we bound the distance between the erformance of the two olicies and by a term roortional to the distance between the broadcast models and (estimation error). As noted earlier, tne key feature of an ad hoc network is that there exists no central control or comutation unit to suervise the imlementation and calculation of routing decisions. This feature underlines the imortance of roviding a distributed algorithm for the comutation and imlementation of an otimal olicy. The authors in [4] and [5] rovide algorithms in which each node comutes its otimal local routing actions in a distributed fashion, i.e. each node only uses the local information available to it to make routing decisions. It is shown that under a technical condition, which holds almost in all real scenarios, these algorithms all converge to an otimal stationary olicy which is consistent with the otimal index olicy comuted centrally. Combining this result with our sensitivity analysis for the centralized control roblem, we extend our sensitivity results to otimal routing strategies that are comuted in a distributed fashion. The remainder of this aer is organized as follows. In Section II, we formulate the roblem we analyze. Secically, in Section II-A we first resent the formulation of the routing roblem in an ad hoc networks, rovide some useful notation and definitions and state the result given by [4] and [5] on the structure of the otimal samle ath deendent routing (SPDR) strategy. In Section II-B, we formulate the roblem of sensitivity of routing olicies with resect to errors in channel estimation and establish the desirable goals of such study. In Section II-B.3, we construct examles to illustrate that such goals may not always be achievable. In Section III we resent the main results of our sensitivity analysis. In Section III-A we establish the mathematical relationshis between loss of erformance and the error in the estimation of the local broadcast models. In Section III-B, we discuss the essence of our sensitivity results through examles and numerically comare the robustness of otimal SPDR with that of other known routing strategies. In Section III-C, we extend our results to the otimal routing olicies which are comuted and constructed in a distributed fashion. In Section IV, we conclude the aer. II. PROBLEM FORMULATION We first revisit the following roblem (Problem ( )) formulated in [4], [5], and [6], and state the results that are necessary for our analysis (for detailed roofs see [5]). Based on Problem ( ), we formulate Problem ( ) which is an investigation of the sensitivity of an otimal routing olicy with resect to errors in estimation of the transmission robabilities which describe by the local broadcast model. A. Problem ( ): Statement and Results 1) Model ( ), Notation, and Preliminaries: We begin by briefly defining notation and stating definitions for the system model, which we refer to as Model ( ), under consideration. As discussed in [4], Model ( ) is robabilistic, and control is centralized, meaning that the controller has access to all the information available at the network. A descrition of the elements of this network model is given below. is the number of nodes in the network. is the set of all nodes. So.

4 4 refers to a state of the system, defined as the set of nodes which have received the message. refers to the state at time. We define. We write to indicate the robability of reaching state from state when choosing node for transmission,. We write as shorthand for. We refer to as the broadcast model. We define. We assume that transmission events at a given node are i.i.d., and transmission events are indeendent among all nodes. Node is called a neighbor of node. Given the local broadcast model, is the set of all neighbors of, together with itself. Note that is ermitted. Definition 1 Increasing Proerty: Model ( ) is said to have the increasing roerty for any system realization under any olicy we have. Definition 2 Decouling Proerty: Model ( ) is said to have the decouling roerty successful transmission from a node to a set of neighbors at a given time is unaffected by which other nodes already have the message. We assume that Model ( ) has both the increasing and decouling roerties. is the reward function, and. Also. is a Markov olicy. We write to indicate olicy transmits at node when in state. We write to indicate olicy retires and receives reward when in state. For convenience we write as shorthand that olicy retires and receives. In this case, we say that olicy retires and receives the reward of node. By, we mean both and. By, we mean either, or, for some. Each transmission from node incurs a cost of. We next formulate the centralized version of the stochastic routing roblem with time-invariant arameters. 2) Statement of Problem ( ): ( ) We consider the transmission of a single message, from a given initial state (i.e. a given set of nodes) to a set of destination states, in a wireless ad hoc network of nodes described by Model ( ) in which the transition robabilities are given by the broadcast model. Transmission instances occur at discrete time oints. Each transmission from a given node incurs a fixed cost. According to Model ( ): ( ) at each transmission instance the transmitting node is chosen by a central controller that always knows the current state of the system (i.e. the set of nodes that have the message); ( ) node transmissions are local broadcasts, that is, multile neighbor nodes may all simultaneously receive the message; ( ) given the node chosen to transmit, the robability that a given set of nodes receives the message is known and fixed; ( ) The central controller is informed, without any cost, as to which nodes receive the message. Control information flow between the nodes and the controller is considered free of energy and instantaneous in time; ( ) each transmission event is assumed indeendent of those before and after; ( ) a reward function is secied. At any instance, the central controller can terminate the transmission rocess or choose to continue transmitting. The objective is to choose: ( ) the node to transmit at each transmission instance, and ( ) the instance to terminate the transmission rocess, to maximize over all Markov olicies, (1)

5 5 where is the transmission/termination olicy the controller follows, is the time when the transmission rocess is terminated under olicy, is the state at, is the node chosen by the transmission olicy at time, and is the exected reward when starting in state under olicy under local broadcast model. Restriction to Markov olicies does not entail any loss of otimality because Problem ( ) is a stochastic control roblem with erfect observations [7]. Mathematically, Problem ( ) is arameterized by a tule, where. 3) The Transmission Control Problem: Consider an ad hoc network in which control of transmission tye (in terms of ower, antenna directionality, and addressing) is allowed. In such a network at each time ste the central controller chooses a node for transmission, among the nodes with the message, and a transmission tye, among a finite set of allowable tyes, is chosen for that node. To each node and transmission tye, a transmission cost and a robability distribution, denoted by, describing the robability that a given set of nodes receive the message are assigned. The objective for the controller in such a network is to determine a olicy which maximizes a total reward similar to that of equation (1). Such a olicy secies the otimal number and coverage of hos, along each realization of the oeration of network. Such a network can be modeled by Model ( ) and can be formulated as Problem ( ) with a articular structure on the robability of successful transmission, i.e. a articular structure on the broadcast model. This is ossible since in Model ( ) no articular assumtion regarding the statistical correlation among the transition robabilities has been made. We seek to construct a articular network described by Model ( ), which satisfies the conditions required by the addition of multile transition tyes resulting from the use of multile ower levels. In order to do so we reresent each node as a set of sister nodes with cardinality, where refers to the number of transition tyes available at node. Each sister node in such a set reresents a transmission tye for node, and is identied by the air, where and. We define to be the collection of these sets of sister nodes. Transmissions in the sace are based on the corresonding events in, as follows. Each transmission at node with transmission tye corresonds to a control decision which chooses node. Such a transmission incurs a cost. If such a transmission leads to a set of nodes, say, receiving the message, all nodes such that receive the message. This imlies that all sister nodes receive the message simultaneously, or in other words, message recetions for sister nodes in are deterministically couled. Finally each sister node of receives the same reward described by the reward function. The roblem of otimal routing in the ad hoc wireless network described by is a secial case of Problem ( ), when transmission robabilities have a articular couled structure to accommodate dferent transmission tyes as sister nodes at which the message recetions are deterministically couled (for more details see [4] and [5]). Hence the sensitivity results derived in this aer aly to a wireless ad hoc network with ower control and multile transmission tyes. 4) Critique of Model ( ) and Problem ( ): In this section we briefly discuss the fundamental assumtions and modeling choices in Problem ( ). We, like the authors in [5], oint out that a model s value deends on how well it catures imortant asects of reality. Furthermore, ad hoc networks vary greatly in their characteristics in terms of mobility, transmission channels, etc. Hence, the validity of the model deends on these characteristics, i.e. a good model for a articular network with low mobility and heavy traffic may be of no use for a heavily dynamic network with rare messages and vice versa. In this section we discuss the fundamental assumtions used in the construction of Model ( ), and their imlications on the utility of the resulting otimal routing strategy. The discussion here is qualitative aimed at roviding insight into the alicability of Model ( ) to various ad hoc networks. Hence no attemt is made to recisely quanty the henomena we address. The first assumtion is the broadcast nature of local transmissions in Model ( ). This assumtion is justied by the omni-directionality of the antennas. It imlies the ability of multile neighbors to receive a message simultaneously, decode its header, and acknowledge the transmitter.

6 6 Model ( ) is based on the existence of a central controller with erfect knowledge of the state of the system. In Section II.A.5, we resent a riority olicy which can be imlemented in a distributed fashion and is otimal for the centralized routing roblem. Such an otimal olicy does not require the central control and information available in Model ( ). Hence, the results of this aer remains valid for the real scenarios where the centralized information and control assumtion is relaxed. Another assumtion is the time-invariant nature of the reward for the recetion of a message at its destination, and a fixed consumtion of transmission energy (can be extended to average recetion) at each node. As a consequence of this assumtion, the objective is to minimize the energy consumtion (dominated by transmissions and recetions), even at the exense of timeliness or system caacity. In Model ( ), each message is considered in isolation. This imlies that the result is valid mainly for systems where in site of the resence of sufficient messages in the system to justy Route Maintenance, the neighbor interference is not the dominant channel imairment. Finally, in Model ( ) it is assumed that transmission events at a given node are described by a time-invariant conditional distribution, which is indeendent among all nodes. We clary the meaning of this assumtion in terms of the channel and network models. Though this assumtion imlies a time homogeneity on transmission robabilities, we can satisfy this requirement with less rigid assumtions such as (1) slow change of toology and (2) restriction of channel degradation to either slow or fast channel effects. 5) Preliminary Results: Otimal Routing in Problem ( ): In this section, we summarize the results in [4] that are relevant to our work. For that matter we need the following definitions: Definition 3: A Markov olicy is a riority olicy there is a strict riority ordering of the nodes s.t. we have or, where is the set of nodes of riority lower than under. Definition 4: For riority olicy, we write when has higher riority than under. Definition 5: For riority olicy and node, we denote by the class of higher riority subsets of neighbors of, i.e.. Similarly we define Now we state the following facts from [4] (for more details see [5]). Fact. 1: For riority olicy we have, when for. This fact holds by the decouling roerty and the definition of a riority olicy. Fact. 2: For riority olicy and any state where or we can write the exected reward as selects transmission selects retirement (2) Fact. 3: There is an otimal Markov olicy for Problem ( ) which is a riority olicy and whose exected reward has the following roerty: (3) Fact. 4: Under the otimal Markov olicy the exected reward for each node defines an index (4) this index, in turn, defines an otimal ordering of the nodes and the actions taken at these nodes; i.e. (5) All these facts which are roved in [4] establish that, under the assumtion that the network arameters are fixed and known, there exists an otimal routing riority ordering of nodes. The authors in [4] roose a centralized algorithm (Algorithm 1 in [5]) with comlexity which comutes the otimal riority listing of nodes in

7 7 Problem ( ) when all the network arameters are known and available at the same location (this algorithm is similar in nature to standard Dijkstra in OSPF [8]). Furthermore, three distributed algorithms to comute the otimal riority listing of the neighbors at each node are roosed in [5]. Note that due to the nature of the otimal riority olicy, there is a natural distributed imlementation of such olicy; such an imlementation requires that (1) successful recetions at neighboring nodes are acknowledged to the transmitting node, and (2) each node kees a riority list of the node itself and its neighbors. A node transmits until another node of higher riority successfully receives the message. Thus, the network layer does not dictate the route, and in fact there is no one route, but the actual route a message takes between source and destination is samle ath deendent. This characteristic of the solution is distinctly dferent from that of other roosed algorithms (see [2], [9], [8], [3], [1]). The above model and all roosed algorithms assume knowledge of the transmission robabilities. To actually imlement the algorithms, methods to estimate these robabilities should be emloyed (see discussion in [5]). On the other hand, such methods introduce various levels of error. In other words, in reality is not known but has to be estimated. The resence of estimation errors raises the imortant issue of the sensitivity of the results in [4], [5], and [6] with resect to (small) variations in. This motivates the sensitivity analysis resented in this aer. Based on Model ( ) and the above reliminary results we roceed with the sensitivity analysis of Problem ( ). B. Sensitivity Analysis: Problem Formulation 1) Statement of Problem ( ): Consider Problem ( ) associated with two sets of system arameters, and, describing the true and estimated models of the system, resectively. According to the results given in Section II-A.5 there exists an index olicy which is an otimal routing olicy for Problem ( ) with arameters. At the same time the otimal solution to the estimated model,, is an index olicy that is not otimal for the true model, in general. Policy is alied to the system with the true broadcast model (distribution). We are interested in: ( ) Determining/quantying the dference between the erformance of olicy in such a system and the best ossible erformance, achieved by. ( ) Relating the aforementioned dference to a quantity describing the estimation error in the (true) broadcast model. To quanty the dference secied in ( ) we define an aroriate metric on the sace of all routing olicies. We define the distance between olicies and at state in the context of the distribution as We define the distance between olicies and in the context of distribution as To relate the dference secied in ( ) to the estimation error in the (true) broadcast model we first quanty this error by defining a distance measure between the true broadcast model and the estimated model. We use the total variation metric for this urose (see [10], [11]). The total variation distance between two local broadcast models, and, describing the robabilities of transmission success for node, is defined as (6) (7) (8) We extend this measure to define the total variation distance between broadcast models and as (9)

8 8 Source 21!c2 23!c1 12 R Destination! Fig. 2. Examle 1. Note that at each node transmissions to dferent neighbors are indeendent and the maximum error on each link is, i.e. for,, then. Hence, the total variation distance is an aroriate metric to secy the estimation errors. Based on the above we formulate the following sensitivity analysis roblem. ( ) Consider Problem ( ) for two sets of arameters, and, describing the true and estimated models of the system, resectively. Let be an otimal routing olicy for Problem ( ) with arameters, and be an otimal routing olicy for Problem ( ) with arameters. The objective is to determine the distance between olicies and in the context of, and relate this distance to the total variation distance between the estimated model and the true system model, i.e.. Routing olicy is said to be robust with resect to errors in channel quality estimates, is bounded by a finite term roortional to error. 2) Basic Assumtion on Convergence Rate of Estimation Algorithm: As mentioned before, the estimated local broadcast model is constructed through an on-line arameter estimation algorithm. Denote by the characteristic time of the estimation algorithm; throughout the aer, by characteristic time of a rocess we mean the time required for the transient behavior of a rocess to settle down. For examle, the characteristic time of the estimation algorithm is the time required for the estimation algorithm to converge to some final estimate. We assume throughout the following analysis that, where is the characteristic time for the comutation and dissemination of the calculation of otimal ranking of the nodes through the network, and is the characteristic time of signicant network toology variation. 3) Examle: The following examle shows that, in general, otimal routing can be extremely sensitive to estimation errors, i.e. there exist scenarios where a small error in estimation can unboundedly deteriorate the erformance of the constructed riority olicy. Consider the simle network given by Fig. 1. We assume that transmission success robabilities are given by the true model and the estimated values of these robabilities are given by model. The value of these transmission robabilities are,,,, and finally. Furthermore, we assume while. Assume that node has transmission costs ;. Rewards are zero for the first two nodes, and it is equal to at the destination. The cost of transmission at node 2 is much larger than the cost of transmission at node 1, i.e.. In this examle the total variation distance between the two broadcast models and, i.e. is. There are two riority olicies and ossible in this network. Under these olicies we have and. The distance between these two olicies in the context of is infinite, since. We show that even for a small distance between models and, i.e. small, olicy can be selected as the otimal olicy (due to its otimality in the context of ). To rove this, we write the exected

9 9 rewards: Since, there exists a (small) for which is selected as the otimal riority olicy when the estimated distribution is assumed to describe the transmission success. On the other hand,. Therefore we have even though. This examle illustrates that in general a small error in channel estimation can cause an unbounded decrease in erformance. A. Analysis of Problem ( ) III. ANALYSIS AND RESULTS In this section our goal is to bound the distance between two olicies and by a term roortional to the distance between the broadcast models and. As illustrated in Examle II-B.3, this is not ossible in general. Thus, to obtain a ositive result, we make the following assumtion on the nature of the estimation error. Assumtion 1: For any node such that, that is, does not retire at node, there exists such that Intuitively Assumtion 1 imlies that the magnitude of the estimation error at each node can not be larger than the total estimated robability of routing the message closer to the destination. In other words, even under the subotimal routing olicy the transmission at each node has a true (according to P) ositive robability of reaching a node of higher riority. Under the above assumtion we seek to bound the loss of erformance by a term roortional to the estimation error. To do so we first resent the following definition which will simly our notation. Definition 6: For any measurable function which is -integrable, we define the Markov oerator (13) Using this definition we can write the exected reward for riority olicy at state, assuming or, as selects transmission (14) selects retirement We begin by establishing Lemma 1 which relates the overall distance between two olicies and to the distance between the olicies at each singleton. Lemma 1:. Proof: Let and. Then, (10) (11) (12) (15)

10 10 The second equality in equation (15) is true since is otimal in the context of, and the first inequality holds because olicy is otimal in the context of. Hence, (16) To bound the erformance loss, we develo uer bounds on each of the maxima that aear in Lemma 1 (right hand side of equation (16)). Bounds on the first term, i.e., are obtained via Lemma 2. Bounds on the second term, i.e., are obtained via Lemmas 3 and 4. The roofs of Lemmas 2 and 3 are given in Aendix III. Lemma 2 below establishes at each node a relationshi between and the distance between broadcast models and. Lemma 2: Assume that is the nodes strict riority ordering under olicy, i.e.. Then, we have (17) where is increasing in and satisfies the recursion (18) Lemma 3 below establishes, at each node, a relationshi between and the distance between the broadcast models and. Lemma 3: Assume that is the nodes strict riority ordering under olicy, i.e.. We have (19) where is increasing in and satisfies the recursion (20) Remark: In Examle II-B.3 we have which imlies the unboundedness of the right hand side of equation (19). This is consistent with the result rovided in Section II-B.3, establishing that. Note that Lemmas 2 and 3 do not assume the validity of Assumtion 1; hence, they can be alied to Examle II-B.3. Lemma 4: Under Assumtion 1, we have where is increasing in and satisfies the recursion (21) Proof: Case 1: If, then. Hence, (22) (23)

11 11 On the other hand, the right hand side of equation (22) is always a ositive number. Hence, (24) This comletes the roof in Case 1. Case 2: If, then Assumtion 1 imlies that: (25) On the other hand we have (26) (27) where the first equality holds since for every,, the third inequality follows the definition of total variation metric, and the last inequality holds because of equation (25). The assertion of the lemma follows from equation (26) and Lemma 3 Combining Lemmas 1, 2, and 4 we establish the following theorem, which summarizes one of the two main results of this aer. Theorem 1: Under Assumtion 1, we have where the function denotes the otimal robabilistic connectivity for a broadcast model, its corresonding otimal riority olicy, and the vector of bounds on estimation error denoted by and is defined as (29) (28)

12 12 Proof: Combining Lemmas 1,2, and 4, we have where the first inequality holds because of Lemma 1, the second inequality is a result of Lemmas 2 and 4, the first equality result from the definition of functions and and their monotonicity in the index, and the last equality follows from the definition of the function. The term in equation (28) deends on the toology of the network under the true and the estimated broadcast models. This deendency rovides insight into the study of sensitivity of the otimal routing olicies with resect to the estimation error under various toological structures. It can be seen that loss of erformance for networks where each transmission reaches a large set of higher riority nodes is smaller than for networks with hot links, where the condition of a few links is critical in determining the ability of the routing olicy to transfer the message to the destination. This is an imortant feature of the roosed otimal riority routing olicy. We will elaborate on this in Section III-B. On the other hand, sensitivity analysis may be used to rovide guidelines in designing on-line estimation algorithms with accetable margin of error. In such alications, the deendency of our bounds on the true and estimated structures and toology of the network can create dficulty. In general and in a ractical setting, such models are not known and cannot be exloited to design aroriate algorithms. Furthermore, in an ad hoc network it is undesirable to assume any articular toological structure. For these alications, we rovide Theorem 2 below to eliminate the deendency of our bound on the articular (and unknown) toology of the network. To do so, we need Lemma 5 and Corollaries 1 and 2 that follow. Lemma 5 is a direct consequence of the results given in [4]. Lemma 5: Let and be two otimal riority routing olicies under broadcast models and, resectively. Assume that, that is, olicies and do not retire when the state is. Then (30) (31) Proof: imlies that (32) On the other hand, for and any olicy we have. Hence, (33) This imlies that. Similarly,,. The roof of Lemma 5 is now comlete. Corollary 1 of Lemma 2: Assume that is the nodes strict riority ordering under olicy. Then, we have (34)

13 13 where is increasing in and satisfies the recursion Proof: The assertion of the corollary follows directly from Lemmas 2 and 5. Corollary 2 of Lemma 4: Under Assumtion 1, we have (35) (36) where is increasing in and satisfies the recursion Proof: The assertion of the corollary follows directly from Lemmas 4 and 5. We now use Corollaries 1 and 2 to rove Theorem 2, which rovides a bound on the error indeendently of the toology. Theorem 2: Under Assumtion 1, we have where Proof: Combining Lemma 1 and Corollaries 2 and 3, we have (37) (38) (39) where the first inequality holds because of Lemma 1, the second inequality is a result of corollaries 1 and 2, and the first and second equalities result from the definition of functions and and their monotonicity in the index. 1) Sufficient Conditions for Assumtion 1 to Hold: Assumtion 1 is the minimum requirement needed to guarantee and obtain a finite bound on the sensitivity of the otimal index routing olicy to the estimation error in the broadcast model. This assumtion deends on the nature of the otimal olicies (under the true model and estimated model ), hence the toology of the network. Nevertheless, there are stronger conditions which are indeendent of the network toology, are sufficient to guarantee Assumtion 1, are easy to very, and are given below. (40)

14 14 Energy Cost of Various Policies in Network Exected Total Cost (e.g. consumed energy) True Otimal SPDR Estimated SPDR OSPF!tye Sensitivity Bounds Network 1 (Only One Route) Percentage of Error in P 23 Fig. 3. Network Condition 1: For any node, there exists such that (41) Condition 1 guarantees that there exists for which Assumtion 1 is satisfied. Intuitively, Condition 1 has two signicant imlications. First, it imlies that the network toology under the estimated broadcast model does not contain links which do not really exist, i.e.,. Second, the condition imlies that, whenever there is a link between nodes and, i.e., there is a finite bound on the ercentage of error in over-estimation of the quality of the link, i.e.. In other words, secies the maximum error ercentage in the estimation of the quality of links connected to node. Condition 2: The estimation error is bounded by the following exression: From Lemma 5 it follows that Condition 2 guarantees that Assumtion 1 holds. Although Condition 2 is too restrictive, it can be checked without any knowledge of the robabilistic toology of the network and/or the structure of the otimal olicies. Intuitively, this is a sufficient condition to guarantee a Lischitz-tye continuity of the erformance, which in turn imlies a bounded gradient around the true model. Unlike Condition 1, Condition 2 bounds the absolute value of error in estimation rather than the error ercentage. B. Examles and Discussion In this section we resent three networks ( ) given by figures 2-4 to illustrate how toology affects the sensitivity of the otimal routing olicy with resect to the estimation error. Consider these three networks. We assume that successful transmissions along dferent links are indeendent. Hence, Model ( ) for network,, can be defined by where is a transition matrix whose -th element reresents the robability of successful transmission from node to. We assume,,,,,, and. Consider routing a message from node 1 to node 5 in networks. We assume that the estimated value of the transmission robabilities are (42) and

15 15 Energy Cost of Various Policies in Network Exected Total Cost (e.g. consumed energy) True Otimal SPDR Estimated SPDR OSPF!tye Sensitivity Bounds Network 2 (Parallel Routes) Percentage of Error in P 23 Fig. 4. Network Furthermore, we assume that the robabilities of successful transmission between each air of nodes are known excet for the link between nodes 2 and 3 in each network where there is an error in the estimation of the quality of the link. Hence, for any error value and, we have, where reresents the true broadcast model for Network, and is a matrix whose only non-zero element is the th element which is equal to 1. Notice that. We vary the error from 0 to 100 ercent of the estimated value of the link s transmission robability, and comare (1) the loss of erformance in the three dferent networks, and (2) the bounds rovided by Theorem 1 in each case. Furthermore, for comarison uroses, we rovide the erformance of a OSPF-tye routing algorithm (see [2], [9], [8], [3], [1]) where some form of shortest route is established as the minimum energy route. In this OSPF-tye routing algorithm a full route with minimum exected energy is identied and set u, and all information between source and destination is transmitted on this fixed route. It can be shown [2] that, under the estimated model, the minimum energy route from node 1 to node 5 in all three networks is constructed as. Note that as the error ercentage at link increases, the true cost of routing via this minimum energy route increases. Before discussing the sensitivity of routing olicies in these examles, we would like to oint out a known advantage of using the samle-ath-deendent routing olicy (SPDR) roosed by [4]. As mentioned before, an imortant feature of the roosed otimal routing olicy given in [4] is the fact that the route a message takes between source and destination deends on the articular realizations of channels and transmission success. In other words, when imlementing the roosed SPDR, the network layer avoids establishing a fixed route. It is known (see [4]) that due to this roerty, in well-connected networks the otimal SPDR olicy shows an advantage in achieving lower exected cost over strategies which set u fixed routes. This is mainly because in well-connected networks, due to the resence of multile alternative routes between source and destination, the otimal SPDR strategy avoids unnecessary retransmissions by choosing the best neighbor among those which have received the message. The grahs rovided by Figs. 2-4 illustrate the aforementioned advantage. We use networks - to discuss the nature of our sensitivity analysis. We have set u these examles to illustrate the effect of the network toology on the sensitivity of the erformance of the otimal SPDR olicy

16 16 Energy Cost of Various Policies in Network Exected Total Cost (e.g. consumed energy) True Otimal SPDR Estimated SPDR OSPF!tye Sensitivity Bounds Network 3 (Alternative Routes) Percentage of Error in P 23 Fig. 5. Network (network has the lowest connectivity, while is the most connected). In each case, we study the erformance of the otimal SPDR strategy when the estimated robabilistic model includes an % estimation error over a articular link on the minimum energy route. In addition, we assume that such error occurs in the form of overestimation. For each network we lot the erformance of olicy, which is the otimal (SPDR) riority olicy associated with the estimated model. We rovide the otimal cost, had the true model been known, as a bench-mark. Furthermore, we lot the bounds rovided by Theorem 2. The results (summarized in Figs. 2-4) can be summarized as follows. Network consists of a single route between the source and the destination, hence it is exected for all olicies to demonstrate identical erformance and robustness. In such a scenario, all routing olicies rely on the failing link for their routing decisions. As the quality of the link decreases all olicies fail to successfully transort the message from the source to the destination. Hence, as the quality of the link dros the cost of routing the message from node 1 to node 5 grows due to the retransmission efforts. Notice that due to the retiring otion the otimal cost, had the true model been known, is slightly below the cost of other olicies. In network there are more than one route from source to destination. This feature of the network has no effect on the erformance of OSPF-tye routing strategies, since such strategies always select the route which, under model, is assumed to be the minimum energy route, i.e.. The otimal riority (SPDR) olicy shows better erformance even in the resence of estimation error. Notice there is still only one route from node 2 to the destination, hence as the error goes to 100%, goes to zero and the right hand side of equation (28) becomes unbounded. In network where there exist alternative routes around the failing link, the otimal SPDR olicy shows a high degree of robustness. In addition to caturing the effect of toology on the sensitivity of the otimal olicy, these examles rovide a rough idea on the tightness of the bounds rovided by the right hand side of equation (28). It can be seen that the obtained bound becomes tighter as increases. C. Distributed Comutation of the Otimal Policy As mentioned in Section I, the key feature of an ad hoc network is the absence of a central control or comutation unit, and this underlines the imortance of roviding distributed algorithms for comutation and imlementation of an otimal routing olicy. The authors in [4] and [5] rovide various algorithms in which each node uses its local information in order to comute a local riority list of its neighbors. The riority list for each node is determined based on the estimates of the otimal exected reward of the node and its neighbors. The local information available at each node consists of the local broadcast model for that node and the udated estimates of its neighbors otimal exected rewards. These estimates are received regularly by the node from its neighbors.

17 17 Each node comutes a new estimate of its own otimal exected reward according to an udate equation secied by the distributed comutation algorithms and it regularly communicates this estimate to its neighbors. To a great extent these algorithms are, in their information structure, similar to the Distributed Bellman-Ford algorithm (see [12], [13]), and can be thought of the samle-ath deendent extensions of the distributed Bellman-Ford algorithm. In all the distributed algorithms resented in [4] and [5], each node uses information only about its own local broadcast model and not the global network model. This imlies that, in ractice, it is sufficient for each node to estimate its broadcast model locally. We are interested in studying the loss in erformance of these algorithms when there are estimation errors at the local broadcast model. In [4] and [5] it is shown that under the roosed distributed algorithms the estimates of the otimal exected reward of each node converge to their true values. Furthermore, it is roved that almost in all ractical scenarios this convergence occurs in finite time. In other words, in almost all ractical cases an otimal local index olicy which is consistent with the otimal index olicy described in Section II-A.5 can be constructed in finite time. This imlies that, given a sufficiently long time horizon, all the algorithms of [4] and [5] demonstrate identical erformance loss in the resence of estimation errors in the broadcast model. More secically, assume that the true system model is as before, and the vector reresents a collection of otimal local routing decisions at nodes at time under distributed algorithm (one of the three algorithm rovided in [4]). Furthermore, assume that each node estimates its local broadcast model,, for each in a local fashion based on all its communications, both control signals and messages, as well as its channel measurements. Construct an overall local broadcast model such that for and we have otherwise Assume that vector, reresents a collection of local routing decisions at nodes at time under distributed algorithm and under the estimated model. For almost all ractical cases there exists a large horizon such that for, where is a stationary olicy that is otimal for the centralized roblem with model. This imlies that, in the context of true model, the erformance of olicy, constructed by distributed algorithm is indeendent of and is equal to the erformance of the stationary olicy. On the other hand, due to the otimality of the distributed algorithm, there exists a finite horizon such that olicy is otimal in the context of the true model ; i.e. its erformance is identical to that of the stationary olicy, where is an otimal index olicy for the centralized roblem with the true model. Hence ast horizon, under any of the distributed algorithms of [4] and [5], the loss in erformance due to estimation errors in the broadcast model is equal to the loss in erformance of olicy. Such a loss has been calculated in Section III-A. And it is shown that under assumtions 2 and 3, the erfomance loss is bounded by a term roortional to the estimation error, i.e. the distance between the true model and estimated model. Theorem 3 summarizes this discussion in a recise fashion under the following assumtions. Assumtion 2: For any node such that, there exists such that This assumtion is identical to Assumtion 1. Assumtion 3: For any air of nodes and such that,. This assumtion imlies that no two neighboring nodes have the same exected reward (i.e. otimal index) to route a message to the destination. It holds in almost all ractical scenarios. Under this assumtion, the olicies constructed by any of the three distributed algorithms in [4] converge to a stationary otimal index olicy in finite time, i.e. for all sufficiently large. (43) (44)

18 18 Theorem 3: If Assumtions 2 and 3 hold and is sufficiently large, then for and for any of the distributed algorithms of [4], say, where the function is defined by equation (29). Proof: Assumtion 3 imlies that for every, we have (see the above discussion and [4]), where is an otimal index olicy for the estimated model, described in Section III-A. On the other hand, for all we have, where is an otimal index olicy for the true broadcast model described in Section III-A (see [4] and [5]). Hence, for where the inequality follows from Theorem 1. Notice that in Theorem 3 is the time to comute and disseminate the local otimal olicies. In other words where is defined in Section II-B.2. In general, deends on the values of,, and algorithm. Therefore, the covergence rate of an algorithm varies with the estimation error. This raises the issue of sensitivity of the convergence rate of various algorithms with resect to estimation errors in the local broadcast model. The answer to this issue combined with the result of Theorem 3 can shed light on the transient behavior of the distributed algorithms roosed by [4] and [5]. Hence, the effect of estimation errors on the convergence rate can be considered an imortant goal of future research. (45) (46) IV. CONCLUSION In this aer we examined the sensitivity of otimal routing olicies in ad hoc wireless networks with resect to estimation errors in channel quality. We considered an ad hoc wireless network where the wireless links from each node to its neighbors are modeled by a robability distribution describing the local broadcast nature of wireless transmissions. These robability distributions are estimated in real-time. We investigated the imact of estimation errors on the erformance of a set of roosed routing olicies. Our results can be used as a guideline to design on-line estimation algorithms with accetable margin of error. At the same time, our results rovide a method to study the effect of the network toology on the robustness of a routing strategy with resect to errors in channel estimation. We rovided a few numerical examles to illustrate such robustness issues. We believe that such results combined with a study of the statistical distribution of tyical link estimation error can rovide a guideline in designing toologies with desirable robustness to estimation errors and link failures. In summary, the roosed sensitivity analysis rovides a bound on the loss of erformance as a result of a time-invariant error in the estimation of the quality of broadcast channels. In general, the estimation error is a dynamic variable whose variation deends on the dynamics of the network toology, the rate at which otimal routes are udated, and the convergence rate and error margin of the on-line estimation algorithm. We rovided intuitive conditions on the time characteristics of the network s toological changes and the rate of convergence of the estimation algorithm, hence the dynamics of the estimation errors, to allow for our sensitivity result to be extended to more realistic cases. We extended our sensitivity analysis to the case of the distributed comutation of the otimal routing olicy, when the distributed algorithms converge to a stationary olicy. A further study should be conducted to investigate the imact of estimation error on the convergence rate of various distributed algorithms. We would like to oint out that in scenarios where changes in the toology and structure of the network are relatively rare, which imlies a sufficiently fast recovery of information and reconstruction of an otimal routing