Queueing in Space : design of Message Ferry Routes in static adhoc networks

We study the concept of Ferry based Wireless Local Area Network (FWLAN), in which a number of isolated nodes are scattered over some area and where communication betwee n a node and the outer world, or communication between the nodes, are made possible via a message ferry. The Ferry has a predetermined cyclic path which collects packets from a no de and delivers packets to it when it is in the vicinity of the node. We use the mathematical theory of polling systems to study the performance of the FWLAN. We consider three different architectures and each of them is mapped to a polling model. The polling disciplines that are needed for modeling the FWLAN involve non-standard variants of gating disciplines. Our goal is to design the routes of the Ferry as well as the points where it should stop to distribute and collect packets. This mathema tical modeling brings another dimension to the classical relatedvehicle routing problem due to the radio channel: the cyclic path of the ferry need not touch every node. The distance between the nod e and the fairy at the point when communication occurs determi nes the transmission rate and hence the service time and thus the system’s capacity. I. I NTRODUCTION Message Ferry are mobile relays or mobile base stations that serve as ”postman” to deliver to static or dynamic wireless nodes messages (or packets) and to collect messages from them. Mobile base stations have been proposed in the context of mobile Ad Hoc Networks [15], of Vehicular Ad-Hoc Networks (Vanets) [11] and of wireless (static) sensor network s [12]. In the UmassDiesel project, computer have been instal led in 30 out of 40 buses and these then serve as Message Ferry to deliver messages to throw boxes (see http://prisms.cs.umass.edu/diesel/). In this paper we are concerned with a message ferry that serves as a mobile access point in a local area network which we call FWLAN (Ferry Wireless LAN); the ferry delivers and collects packets from static nodes on some geographic area ∆. This problem is close in nature to the well studied problem in logistics where a central point receives service request s from points on the plain. A service vehicle is then sent for handling the requests and one has to design efficient or even optimal vehicle’s route so as to serve all the points, see [3] . Beyond the resemblance of the vehicle routing problem to the problem of designing a route for a message ferry, we notice also a fundamental difference. In the vehicle routin g problem, the routes need to pass through all the points in space that require service. In contrast, the routes of the message ferry need only pass in the vicinity of nodes (in thei r transmission and reception range). Moreover, this range it self is flexible: assuming a fixed transmission power, the range ca n be increased at the cost of decreasing throughput. The relat ion between the range and the throughput are determined by the radio propagation conditions. Taking into account the above radio conditions, we are concerned with the design of a cyclic route of the ferry and of the location of stops along the route. It is only when reachin g a stop that the ferry collects (uplink) and dumps (downlink) the data from/to all the nodes who are closer to that stop than to other ones. The larger the number of stops, the more time the ferry has to spend for stopping and accelerating. Th e lesser the number of stops is, the larger are the distances at which it has to receive/transmit the data from/to the nodes a nd hence the larger is the time to achieve reliable communicati on. In addition to this trade-off which appears both in the one dimensional and the two dimensional cases, there is another important design issue that is specific to the plane: in order to achieve smaller service times the routes need to be longer , which may increase waiting times. Larger service times that would be needed if the routes were shorter imply smaller achievable throughput of the system. We consider three architectures: • Sensor Access Network (SAN):there is a fixed base station (BS) that is connected to the global Internet (or to other base stations) and thus enables communication between nodes in the FWLAN and the outer world. The ferry brings all traffic from (respectively to) nodes in the FWLAN to (resp. from) the BS. There is no traffic from nodes of the FWLAN to other nodes in the FWLAN. • Hybrid Access Network (HAN): Same as SAN but traffic sent by a node in the FWLAN can also be destinated to another node in the FWLAN. In that case the ferry first brings all uplink packets to the BS and then receives from the BS all downlink packets received during the last cycle including those just brought by the shuttle destinated to other nodes in ∆. Using this architecture we can achieve routing within the area also, however it always takes two cycles to complete the data transfer. • Autonomous Network (AUN): the Ferry serves as a local mobile base station. Thus a packet sent by a node is transferred directly to the destination node without first transmitting through a fixed BS. In this case, in contrast to the HAN architecture, the data routing can take place faster: depending on the location of the source destination nodes and the direction of the ferry’s route, a packet may arrive at the destination in the same cycle of the Ferry. This kind of system can best be studied using a polling system, wherein a ferry serves a finite number of queues in a cyclical order [2], [6], [4], [9], [14], [13]. Two queues ar e considered at each stop. The uplink arrivals from nodes that are nearest to the stop under consideration are modeled as on e queue (uplink queue) while all the nearest downlink arrival s are modeled as the downlink queue. The polling disciplines that are needed for modeling the FWLAN involve non-standard variants of gating disciplines . In SAN architecture, we note that upon arriving at a queue, th e ferry serves (bring to the queue) all packets that were prese nt at the base station when the ferry last visited the base stati on. If only downlink traffic existed then this would correspond to the ”globally gated” (GG) discipline [6] (otherwise we call the discipline PGG for ”partially globally gated”). In contrast, when the ferry arrives at a stop it can upload all th e traffic present there upon arrival (so that the standard gate d or exhaustive disciplines can be used to model this). In the autonomous network case, the polling discipline in the upli nk queues will be shown to be a complex combination of PGG disciplines, related to the models in [9], [13]. In designing the ferry’s routes and stops, we aim at minimizing the expected virtual workload in the system (which is some appropriate weighted expected waiting time (WWT) of a random customer). The system, model and the notations of the paper are introduced in Section II. We consider SAN, HAN, AUN architectures respectively in Sections III, VI and V. The theoretical results obtained were simulated using some numerical examples in the respective sections itself. The pap er is concluded in Section VII. II. SYSTEM MODEL AND NOTATIONS We consider a 1-Dimensional or 2-Dimensional geographical area∆ in which static nodes (sensors) are scattered. We assume that the network is sparse and there is no direct globa l connectivity. In order to receive messages (which we call al so ”packets”) from the nodes or to send messages to them, a ferry called ”message ferry” or ”message shuttle” moves around and serves as a postman. The nodes either generate data to or require data form nodes within and/or outside the area. In order to route the data to and from outside the area, the shutt le has to pass through a base station that serves as a gateway. It is possible that the BS is also needed to route the data within the area (for example in SAN architecture). In Sectio n III on SAN architecture, all the data routing takes place via the BS: the ferry goes to the BS once in every cycle to collect and deposit the information. The ferry ”serves” the nodes at each stop in a cyclic manner. We use throughout terms from queuing theory; by ”serves” a message we mean that the ferry transmits it if the connection is downlink (i.e. the message is destinated to a node), or receives it, if it is an uplink messa ge. In Section V on AUN architecture, we consider instead the situation where the server/ferry routes the data directly t o the destination, if an uploaded packet was meant for a node withi n the FWLAN. Only the packets from/to the nodes outside ∆ are routed to BS. Finally in Section VI, we give a brief idea to study the HAN architecture. Ferry’s Route : The ferry moves in a closed path repeatedly and stops at the same finite number ( σ) of predetermined stops in every cycle. The area is divided into σ disjoint subareas and each stop is associated with one of the subareas . A node belongs to that subarea if and only if its signal is strongest there. At each stop, the ferry serves all the nodes located in the associated subarea. Let {Q1, Q2, · · · , Qσ} represent the location of stops of the ferry. For eachi let Ii represent the subarea associated with stop i. We assume that BS is located nearQ1. The indexing in this paper is done in a circular manner. Arrival process : We consider traffic generated at the nodes which we call ”uplink”, and traffic that arrives to the nodes which is called ”downlink”. We shall use for both cases the term ”arrival”. Uplink/Downlink traffic arrives accord ing to an independent marked point processes {Tn, Mn}, where Tn is the arrival time of thenth point andMn = [Ln, L d n, ηn] are the corresponding i.i.d. marks: • Tn is a Poisson point process with parameter λ, • Ln is the source of the uplink (upload to Ferry) while L d n is its downlink (download from Ferry) node. Both Ln, L d n are in∆. When the BS is involved in data transfer then we either consider Ln or L d n appropriately to be the BS. This for example occurs whenever