ANALYTICAL MODEL OF A VIRTUAL BACKBONE STABILITY IN MOBILE ENVIRONMENT

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1 (The 4th New York Metro Area Networking Workshop, New York City, Sept. 2004) ANALYTICAL MODEL OF A VIRTUAL BACKBONE STABILITY IN MOBILE ENVIRONMENT Ibrahim Hökelek 1, Mariusz A. Fecko 2, M. Ümit Uyar 1 1 Electrical Engineering Dept., CCNY, The City University of New York, NY, USA 2 Applied Research Area, Telcordia Technologies, Inc., Piscataway, NJ, USA 1 INTRODUCTION The reliable server pooling (RSP) [4] provides a client s application (also called Pool User (PU)) with a range of reliability services, from server selection to an automatic sessionfailover capability. Servers with the same application functionality, which are also called Pool Elements (PEs), are grouped into server pools identified by a pool handle. A PU accesses the PEs by querying its Primary Name Server (PNS). Recently, we have defined, implemented, and demonstrated a new RSP architecture called Dynamic Survivable Resource Pooling (DSRP) [1], which deploys NSs on a dynamic virtual backbone (VB) for ad hoc networks. The DSRP architecture consists of two independent parts: (1) formation and maintenance of the backbone, where the most stable nodes are dynamically selected as the backbone nodes, and (2) distribution of resource registrations, requests, and replies over the mesh of backbone nodes. This paper focuses on modeling the first part of the DSRP. The base model employs the dynamics of nodes (NS, PE, PU) and VB driven by a node movement, where link creation/failure is modeled via a random walk with probabilistic state-transition matrix [3]. Because the backbone formation algorithm gives the preference to nodes with the small number of link changes and the high degree, the link arrivals and departures determine the probability (and thus the expected time) for an NS to leave, join, or remain in the backbone, i.e., the stability of a dynamic structure of NSs. We developed an initial model for the DSRP to calculate the following end-user metric: What is the expected delay to get service request resolved? The stability and accessibility of a virtual backbone is a major contributing factor to this delay. We thus obtain several related metrics: the probability of a PE/PU not having an operational PNS; the stability of an NS, Prepared through collaborative participation in the Communications & Networks Consortium sponsored by the U.S. Army Research Lab under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. Copyright c 2004 Telcordia Technologies, Inc. and City University of New York. All rights reserved. i.e., the expected time for an NS to leave the backbone; and the expected delay for a PE/PU to find another PNS when the previous one becomes unavailable. 2 BASE MODEL In our approach, we construct a state transition matrix M so that each element M i,j could represent the probability to transit from the i-th to the j-th state within one time unit. When we use the steady state probabilities for M, we can derive a new Markov chain whose states represent the number of available links for a specific node. There may be multiple link arrivals and multiple link departures in one time unit, characterized by a pair of random variables. Based on these random variables, we obtain the stochastic point process that defines the probabilities of a node s degree to drop below or exceed the threshold as the node moves. The expected time of this event determines when an NS turns into a non-backbone node (and vice versa). Ref. [3] uses the discrete-time random walk model to predict the route life time in multi-hop mobile ad hoc networks. Two mobile stations can move in one unit time, and a vector representing a wireless link between two mobile stations is called a state. For example, if one two nodes are in locations (0,0) and (x,y), respectively, then link state between these nodes is < x, y > (< x 0, y 0 >). A random walk model is applied to formulate how a wireless link changes states. Consider any wireless link < x, y > connecting two mobile nodes: after one time unit each mobile node moves into one of its six neighbor cells with probability of 1/6 for each direction. Therefore, there are 36 possible combinations for the next link state. A state reduction technique decreases this number to only 19 possible combinations for the next link state. Finally, a state transition matrix M such that each element M i,j represents the probability to transit from the ith state to the jth state is constructed. Matrix M represents the state transition probabilities after one time unit. Similar to Ref. [3], we further partition cells into layers, where cell (0,0) is on layer 0, the six cells surrounding cell (0,0) are on layer 1, and the outer cells surrounding cells at layer i are on layer i + 1. We followed the procedure in 1

2 Ref. [3] to construct matrix M. However, in our case, M contains more outer states because we have to consider all possible available and unavailable links that may become available. Moreover, to approximate a limited geographic area, our M matrix is constructed in such a way that outer states in the last two layers bounce back to both the inner and the outer states. 3 ANALYTICAL MODEL For N mobile nodes and a given link with state i, which can be either available or unavailable, let P a (i) and P u (i) denote the probabilities that the state will be available or unavailable in the next time unit, respectively. Given k available links, we want to calculate the probability P k,k+1 that there will be k+1 available links in the next time unit. This is possible only if l links disappear and l+1 links appear in the next time unit for l = 0, 1, 2,..., k. Let P dap (k, l) denote the probability that l of k available links will disappear and P ap (K u, l + 1) denotes the probability that l+1 of K u unavailable links will appear in one time unit. If we use the steady state values of the state transition matrix M i,j, then P a (i) and P u (i) will be the same for each link state i without depending on the initial link states. Also, for some initial outer (i.e., link unavailable) link states, the probability of going from an outer state to all inner (i.e., link available) states in one time unit is zero when P a (i) and P u (i) are used without the steady state values (P a (i) = 0 for some outer link states i). However, there may be a certain probability that a link in an outer state will be in an inner state after a certain time. When we use the stationary distribution of M i,j, this probability will be accounted for in the model. Then, the steady state values of P a (i) and P u (i) are calculated as follows: P u (i) = P a (i) = S T j=s a+1 s a j=0 M i,j = 1 s a j=0 M i,j = 1 P a = P u (1) M i,j = P a = 1 P u (2) where j = 0, 1, 2,...., s a represent available link states (inner states) and j = s a +1, s a +2,...., S T represent unavailable link states (outer states). When we use the steady state probabilities of the M matrix, then P dap (k, l) and P ap (K u, l + 1) are as follows: ( ) k P dap (k, l) = Pu l P k l P ap (K u, l + 1) = l ( Ku l + 1 ) P l+1 a a (3) P Ku l 1 u (4) where P a and P u denotes the steady state values of the probabilities that a given link will be available and unavailable at the steady state, respectively. Then, P k,k+1 can be calculated as follows: P k,k+1 = P dap (k, l) P ap (K u, l + 1) (5) The above formulation represents only the probability for a transition from (k)th state to (k+1)th state. However, there may be a transition from (k)th state to (k+h)th state in one time unit for any possible k and 0 k + h K. Given k available links, we want to calculate the probability that there will be k+h available links in the next time unit. This is possible only if l of k available links disappears and l+h of K u unavailable links appears in the next time unit for l = 0, 1, 2,..., k and 0 l + h K u. Let P ap (K u, l + h) denote the probability that l+h links will appear in one time unit. P dap (k, l) has been formulated above. If we use the steady state values of P a (i) = P a and P u (i) = P u, then P ap (K u, l + h) = ( Ku l + h ) P l+h a P Ku l h u (6) Given k available links, let P k,k+h denote the probability that there will be k+h available links in the next time unit, where 0 k K and 0 k + h K. P k,k+h = P dap (k, l) P ap (K u, l + h) (7) By exploiting Eqs. (3) and (6) in Eq. (7), we obtain: P k,k+h = ( ) ( ) k Ku Pa k+h Pu Ku h (8) l l + h where 0 k K, 0 k + h K, and 0 l + h K u We then obtain a new finite state Markov chain using the stationary distribution of the state transition matrix M i,j. In the new chain, there is a transition probability from any state to other states: for each k = 0, 1, 2,..., K and k + h = 0, 1, 2,..., K, there exists a certain probability of going from (k)th state to (k+h)th state, P k,k+h. Let us denote this new Markov chain M i,j. Since this Markov chain is ergodic (finite, connected, and aperiodic), there is a stationary distribution for this process: π k = P r(state = k) for k = 0, 1, 2,..., K. 2 Our aim is now to find the probability mass function for the number of link changes in one time unit. In other words, the new process describes multiple link arrivals and multiple

3 link departures within one time unit. The number of link changes is equal to the number of link arrivals minus the number of link departures, both in one time unit. We want to calculate the probability mass function of the number of link changes, which can be both positive or negative. Let us define a random variable Z that denotes the number of link changes in one time unit. The probability distribution of Z can be calculated as follows: p z (l) = P r(z = l) = K P (k, k + l) π k (9) k=0 where l is an integer between -K and K. We can now obtain the probability mass function for the number of link changes for a node in one time unit. The actual metric that we would like to calculate is the expected time T that a non-ns node will be an NS node or vice versa. Let us concentrate on the former case since the latter case can be found using the similar procedure. Let us define a new set of random variables Z 1,..., Z m that represent the number of link changes for 1 st, 2 nd,..., m th step, respectively. Then the total net number of link changes from the initial time to m-th step will be the sum of the link changes occurred in each step. Let us define a new random variable S m which denotes the total net number of link changes until the mth step which the degree of a node will be equal or greater than the threshold degree: S m = Z 1 + Z Z m. Variable thr 0 represents the difference between the threshold number of available links and the initial number of available links: thr 0 = d thr d 0. We first calculate the probability A m that the degree of a node will be equal or greater than the threshold for the first time, assuming that this event will happen at the m-th step. This means that the degree of this node will be less than the threshold at all steps before the mth step and equal or greater than the threshold at the mth step. Having derived A m, we then obtain the sought metric T : A m = T = m P r(s i thr 0 ( j [0, i])s j < thr 0 ) (10) i=1 m A m (11) m=0 3.1 First passage time analysis for nonns-to-ns case The original problem is to calculate the expected time that a non-ns node, which has initially a smaller number of available links (d 0 ) than the threshold (d thr ), will become an NS. This is possible only when the number of available links for this node will be equal to or greater than the threshold. 3 Table 1: Transition matrix Q: NonNs-to-NS case Q(i, j) = P (i, j) if i < d thr, j < d thr Q(i, d thr ) = K j=d thr P (i, j) if i < d thr, j d thr Q(d thr, j) = 0 if i d thr, j < d thr Q(d thr, d thr ) = 1 Table 2: Transition matrix Q: NS-to-nonNS case Q(i, j) = P (i, j) if i > d thr, j > d thr Q(i, d thr ) = d thr j=0 P (i, j) if i > d thr, j d thr Q(d thr, j) = 0 if i d thr, j > d thr Q(d thr, d thr ) = 1 The above metric T can be obtained as the solution to the Eq. (11), for which we use the first passage time analysis. The first passage time analysis allows us to find the number of transitions made by the process in going from one state to another for the first time. The expected first passage time is the expected number of transitions made by the process in going from one state to another for the first time. Typically, the first passage time analysis is performed for a pair of states. However, in our case, given that a node has the smaller number of available links than the threshold (state i where i = d 0 < d thr ), we want to find the expected first time that the number of available links will be equal to (j = d thr ) or greater than the threshold (j = d thr +1, d thr +2,...., K). In other words, we want to calculate the expected first time that the number of link changes (arrivals) will be equal or exceed a certain number (thr 0 = d thr d 0 ). In the original chain, we combined the states that represent the number of available links equal to or greater than the threshold into a single state (d thr ). We also found the entries of the new transition matrix as shown in Table 1. This new state transition matrix Q is a (d thr + 1) (d thr + 1) matrix and the corresponding Markov chain has d thr + 1 different states. Let us define a new set of random variables X 0, X 1, X 2..., X m that represent the number of available links at initial, 1 st, 2 nd,..., m th time units, respectively. The number of transitions made by the process in going from state i to j for the first time is Let T = min{m 1 : X m = j X 0 = i} (12) denote the probability that T = m. Then f (1) = q (1) = q, = q ik f (m 1) kj (13) k j 0, 1 (14)

4 where q is the element of the ith row and jth column of Q matrix, which denotes the state transition probability of going from state i to j in one time unit. Then the expected first passage time can be calculated as follows: µ = m if if < 1 = First passage time analysis for NS-to-nonNS case (15) All the steps applied in the previous section (nonns-to-ns) are valid in this case. The only differences are that we modify the P and Q matrices as shown in Table 2. 4 NUMERICAL RESULTS In our calculations, the total number of layers (n tot ) is 20 and the number of layers representing the available (inner) link states (n av ) is 5. The total number of layers gives us the maximum distance in terms of hexagonal cells between two mobile nodes in the network. If a distance between two nodes is less than or equal to 5 layers, we assume that these two nodes can communicate with each other through an available wireless link. Each layer may have the different number of link states. For example, layer 0 has only one state < 0, 0 >, layer 1 has only one state < 1, 0 >, layer 2 has two states < 2, 0 > and < 1, 1 >, and layer 20 has 11 states < 20, 0 >, < 19, 1 >,..., and < 10, 10 >. The total number of states is 121 for all layers (n tot = 20). The total number of inner states is 12 for five inner layers (n av = 5) and thus the total number of outer states is 109 (121-12=109). In our calculations, the number of mobile nodes is 106 so that there are 5,565 total bi-directional links in the network and 105 bi-directional links for a single node. These links can be available or unavailable. As numerical results, we report the expected time that a non- NS node becomes a NS (T NS (n tot, n av )), the expected time that a NS node becomes a non-ns (T nonns (n tot, n av )) and the mean number of neighbors (N(n tot, n av )) for a particular node. The condition for being a NS is to have a number of neighbors which is equal to or greater than the threshold and the condition for being a non-ns is to have a number of neighbors which is less than the threshold. The expected times and the mean number of neighbors are function of n tot and n av for a fixed N. 4.1 The expected first times from nonns to NS The expected first passage times (in time units) are shown in Table 3. Table 3: Expected Times for the nonns-to-ns case d 0 d thr T NS (20, 5) By examining Table 3, it is clear that the expected times from any initial state to a certain threshold state are independent of the initial state (the initial number of the available links). This result is expected since we used the steady state probabilities for the availability and unavailability of a given link. A given link without depending on its initial link state (available or unavailable initially) will be available with the probability P a and unavailable with the probability P u in the next time unit. Therefore, the state transition matrix Q has the identical elements at each column, and for a given threshold, the probability mass functions of the first passage times from all states to this threshold state are the same. Since the expected times from any initial state to a certain threshold are independent of the initial state (i.e., the initial number of the available links) as shown in Table 3, we set the initial number of available links to 0 and obtained the expected times for different threshold values. Various cases for different values of parameters are analyzed below. We vary the number of mobile nodes N, the total number of layers n tot, and the number of available layers n av. For all numerical results, N is fixed to 106 and n tot and n av are varied to obtain the different dense networks. Here, the density of a network is defined as the number of mobile nodes per a hexagonal cell. The physical area of a network is determined by the total number of layers n tot. Since we fixed N as 106 and n av as 5 for all numerical results, the density of a network depends only on n tot. We assume that the size of a hexagonal cell is the same for all cases. For n tot = 20, we define the network as medium in terms of its density. We define the network as sparse for n tot = 30, sparsest for n tot = 40, dense for n tot = 15, and densest for n tot = 10. Below we present numerical results for different cases, with the comparison of expected times shown in Fig. 1. 4

5 4.1.1 Medium Network I (n tot = 20 and n av = 5) When the threshold is set to one, there are only two states (0,1) that represent the number of available links for a particular node. Since the initial state is zero, at least one unit time is needed for going from state 0 to state 1 and some expected time ( ). The latter component (0.0002) comes from non-zero probabilities of staying at state 0 in next time units. When the threshold is set to two, there are three possible states (0,1,2). The probabilities of staying at some states (0 or 1) before going into state 2 are higher in this case, and therefore the expected time is higher compared to the case where the threshold is one. From the P matrix, N(20, 5) is calculated as We observed that when the threshold is less than or around 10, the expected first times are small. This observation indicates that the process has a tendency to fluctuate around the expected values. However, when the threshold is greater than 10, the expected first times increase exponentially Sparsest Network (n tot = 40 and n av = 5) We used the total number of layers as 40 and the number of layers that represents the available (inner) link states as 5. The probability of being available in the next time unit at the steady state P a will be much lower than the P a values of the medium and densest networks. From the P matrix, N(40, 5) is calculated as 2.08 compared to N(20, 5) and N(10, 5) calculated as 8.33 and 34.90, respectively. Therefore, it is expected that T NS (40, 5) will be greater than T NS (20, 5) and T NS (10, 5). Moreover, we could not obtain the expected first times for the threshold values greater than 11. Since N(40, 5) is about 2, it takes much more time to visit the states greater than 11. Expected time 10 6 The expected first times vs threshold densest dense normal sparse sparsest threshold Figure 1: The expected times vs threshold to become an NS in different density networks. 5 Future Work The presented work allows the modeling of ad hoc network schemes that depend on either the degree of a node or the rate of change in the set of node s links. Thus far we have focused on the former, but the model can easily be extended to calculate the expected stability of the node, e.g., the number of links transitioning between the states of available and unavailable. This extension should help model the DSRP more accurately (i.e., the stability of a node s links is one of the criteria for joing the backbone), but also to analyze the convergence of routing protocols or bandwidth estimation techniques that depend on the link stability Densest Network (n tot = 10 and n av = 5) We used the total number of layers as 10 and the number of layers which represents the available (inner) link states as 5. The probability of being available in the next time unit at the steady state P a will be much higher than the P a values of the medium and sparsest networks. From the P matrix, N(10, 5) is calculated as compared to N(20, 5) and N(40, 5) calculated as 8.33 and 2.08, respectively. Therefore, T NS (10, 5) is much lower than T NS (20, 5) and T NS (40, 5). Moreover, the expected first times are very close to the ones for the threshold values lower than N(10, 5) = The one unit time is necessary for going from the initial state to the next state and the probability of going into the threshold values is very high, which can be observed from the P a value and the number of nodes in the network. N(10, 5) is about 35, therefore it is expected that it will take much less time to visit the states than References [1] M.A. Fecko, U.C. Kozat, S. Samtani, M.U. Uyar, and I. Hökelek. Dynamic survivable resource pooling in mobile ad hoc networks. In Proc. IEEE Int l Symp. Comput. Commun. (ISCC), Alexandria, Egypt, [2] U.C. Kozat and L. Tassiulas. Service discovery in mobile ad hoc networks: An overall perspective on architectural choices and network layer support issues. [Elsevier] Ad-Hoc Netw. 2(1), pp , [3] Y.-C. Tseng, Y.-F. Li, and Y.-C. Chang. On route lifetime in multihop mobile ad hoc networks. IEEE Trans. Mob. Comput. 2(4), pp , [4] M.U. Uyar, J. Zheng, M.A. Fecko, S. Samtani, and P.T. Conrad. Evaluation of architectures for reliable server pooling in wired and wireless environments. In Li et al., eds, Recent Advances in Service Overlay Networks (S.I.), IEEE J. Select. Areas Commun. 22(1), pp The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Lab or the U.S. Government.