Stability Analysis of Routing Strategies for the Maximum Lifetime Problem in One-Dimensional Ad-Hoc Wireless Networks

Abstract

We solve the maximum lifetime problem for a one-dimensional, regular ad-hoc wireless network with one data collector L N for any data transmission cost energy matrix which elements E i,j are superadditive functions, i.e., satisfy the inequality ∀ i≤j≤k E i,j +E j,k ≤ E i,k . We analyze stability of the solution under modification of two sets of parameters, the amount of data Q i , i ∈ [1, N ] generated by each node and location of the nodes x i in the network. We assume, that the data transmission cost energy matrix E i,j is a function of a distance between network nodes and thus the change of the node location causes change of E i,j . We say, that a solution q(t 0 ) of the maximum network lifetime problem is stable under modification of a given parameter t 0 in the stability region U (t 0 ), if the data flow matrix q(t) is a solution of the problem for any t ∈ U (t 0 ). In the paper we estimate the size of the stability region U (Q 0 , d 0 ) for the solution of the maximum network lifetime problem for the L N network in the neighborhoods of the pointsdescribes the shift of the nodes from their initial locationpartition', the second metric discussed in [3]. In this paper we utilize the fifth metric defined in [3], and we will minimize the maximum node costFor a one-dimensional, ad-hoc networks the metrics 'time to network partition' and 'the maximum node cost' are equivalent because the cutset, a set of nodes the removal of which will cause the network to partition, [4], consists of all nodes of the network. This means that obtained results in this paper are valid for network lifetime problems discussed in [1] and [2]. General formulation of the problem of extending the network lifetime depends on a number of initial parameters which characterize the nodes activity, available resources, the network environment or topology. It is known, that solutions of the problem are very sensitive under change of these parameters, [2]. In general even, if we know an exact solution of a given optimization problem there is a need to investigate its stability under change of these initial parameters to understand the structure of the solution and limitations of its applications, [5], [6]. In wireless ad-hoc networks which characterize limited power of energy, for example in sensor networks, the nodes most of their energy utilize for data transmission and extension of the network lifetime is achieved by designing energy efficient transmission protocols. Knowledge about stability of the exact solution of the maximum network lifetime problem allows to understand limitations of implemented transmission protocols in such networks.In the paper we are considering stability of a solution of the maximum network lifetime problem for a one-dimensional, regular ad-hoc wireless network with one data collector L N . The nodes of the regular network L N are located at the points x i = i, i ∈ [0, N ] of a half-line, where at the point x 0 = 0 there is located a data collector. We investigate stability of the solution under changes of two sets of...

Maximum Lifetime Problem in Sensor Networks with Limited Channel Capacity

“…We show, that for such model the solutions of the maximum lifetime problem for both energy consumption models (1) and ( 7) can be written in the same form and thus are equivalent. We give the solution of the maximum lifetime problem with the new SINR function in the one dimensional regular sensor network L N for both the multipoint-to-multipoint and broadcast data transmission services and show that these solutions coincide with the analytical solution of the problems discussed in [1,8,9]. We also show, that when the distance between sensor nodes is sufficiently large, the solution of the maximum network lifetime problem with Gupta-Kumar SINR function (3) coincides with the solution of the problem written in terms of E i,j and q i,j matrices discussed in [1,8,9].…”

“…In the next sections we show, that the solutions of the maximum network lifetime problem with the channel capacity (15) for the multipoint-to-multipoint and the broadcast data transmission services for both energy consumption models (1) and ( 7) are equivalent. We show that, the solution of the maximum network lifetime problem with maximum transmission rate (15) for both types of services in the one dimensional regular sensor network L N can be obtained from the analytical solutions of the problems given in [1,8,9].…”

“…The maximum network lifetime problem for the multipoint-to-multipoint data transmission service with the channel capacity given by ( 15) is defined by the objective function (8) in which the energy E i of each node is given by the formula…”

“…(21) Proof. It is easy to show that the formulas given in (18) by a simple transformation q i,j → P 0 t i,j and E i,j → γ −1 i,j and Q i → P 0 C 0 Q i can be obtained from the formulas ( 8) from [8], and the solution (21) can be obtained by these transformation from the solution (16) given in [8]. Detailed proof that the set of trees and weight polynomials (21) in the Lemma 1 is optimal can be carried out analogously to the proof given in [8].…”

“…We show, that for a modified SINR function (3), such that the signal of the transmitting node depends on the distance to the receiver, the solution of the maximum lifetime problem for both energy consumption models coincide. We also show, that for a sufficiently large network the solution of the maximum lifetime problem with the SINR function given by (3) can be reduced to the solution of the problem discussed in the papers [1,7,8,9], where the energy consumption model (7) was utilized.…”