In this chapter, we have extended the deployment strategy from Chapter 4, where we have added a replenishment controller in order to extend the lifetime of an initial deployment to meet a mission lifetime. The use of a replenishment strategy is justified by the fact that, as we add additional nodes to the initial deployment, the increase in the lifetime of the network begins to flatten out. Moreover, the quality of the network, in terms of capacity and interference, begins to decrease. We have shown that the problem of replenishing networks with differential node densities is similar to the problem of multi-location inventory control. We showed that the formulation of this type of problem requires a dynamic program with a large state space, and that the non-stationarity of the failure rates further increases the complexity of the problem. Using a technique introduced by Zipkin, we reduced the problem size to a single dimension. The technique requires us to apply the allocation assumption, which holds that, given an order arriving from L periods in the past, the order size
will be sufficient to replenish each location in such a way that the probability that any of the bands will fail over the next lead-time is roughly equal. This assumption holds for our problem the initial deployment and the target levels for replenishment are obtained from the deployment strategy from chapter 4. The initial deployment strategy specifies a differential deployment that aims to balance the residual energy in each band so that a minimum amount of energy is wasted at the end of the network. Another reason that the allocation assumption holds in our case is the way that nodes are allocated to bands in the myopic allocation strategy. The myopic allocation considers both the deviation at the current time period from band target levels and the fraction of failures over the past lag time in each band.
30 40 50 60 70 80 Period 0 5 10 15 20 25 N u mb e r o f F a ilu re s
Figure 5.9: An example of phase transitions in the node failure rate for a single band.
Figure 5.9 shows the cumulative failures of a single band over time. The graph exhibits what appears to be three phases of failure (each phase is boxed in the figure): (1) an initial phase, where no nodes are failing, (2) a “linear” phase, where nodes
begin failing at an approximately constant rate, and (3) a critical phase, where nodes are failing with an exponentially increasing rate. The interpretation of the transition between the linear and critical phases is that, at the beginning of the critical phase, the number of nodes in the band is near the minimum levels set for connectivity guarantees; thus, messages sent from a CH must traverse many more hops to exit the band, and so more energy is being consumed in the band. If bands were allowed to enter this critical phase, the exponential smoothing method used to forecast failures would not be able to track the failure trend accurately, insufficient nodes would be ordered to balance the bands, and the network would fail prematurely. By considering the fraction of failures as well as the target level deviations, we are reacting to both the long-run failures and the recent changes in the failure trends. The result is that the myopic allocation helps to prevent bands from exiting the linear phase.
We tested the replenisher for various lead-time values. The effect of a longer lead- time appears in Figure 5.7(d) for L=20. In this case the lead-time is equal to the planning horizon. We can see from the figure that, in this case, the replenisher consis- tently under-ordered, resulting in node levels well below the targets. A trend between the lead-time and the amounts that node levels fall below the targets becomes clear as we look at Figures 5.7(a)-5.7(d), which show the number of active nodes over time for L = 5, 10, 15, and 20. Since a penalty is incurred for node levels below the target in each period, the effects of the lead-time can be easily seen by comparing the total cost (penalties and actual costs) of a replenishment versus the lead-time as shown in Figure 5.10.
In Section 5.3 we derived a cost function gt, which represents the minimum total
expected ordering and penalty costs incurred in periods t through T . In our im- plementation, gt is computed in each period over the planning horizon, since the
0 5 10 15 20 Leadtime (periods) 10 20 30 40 50 C o st (x1 0 0 0 )
Figure 5.10: The effect of the lead-time on the total cost of the mission.
of the mission lifetime. The value of yt (the order) that minimizes gt is the optimal
batch size for period t. Figure 5.11 shows two instances of gt from our simulations.
The dashed line represents the case where the optimal batch size is zero, thus no order is placed. The solid line represents a case where the minimum value for gt cor-
responds to yt = 477. The jump in the cost functions at yt = 0 corresponds to the
fixed cost for ordering and deploying the nodes. If the fixed cost is increased, the time that an order is placed will be delayed, and the size of the order will increase. The values of the penalty costs d and h will also affect the frequency of orders and the size of the batches. Therefore, given a fixed lead-time, it may be possible to simulate a network prior to deployment in order to find penalty values that will result in better performance.
0 200 400 600 800 1000 1200
Number of Nodes to Order
0.00 20.00 40.00 60.00 80.00 T o ta l C o st Order Nodes Do Not Order Min at 477 Min at 0
Figure 5.11:Characteristic shape of the cost function, gt. The minimum of the cost function
specifies the number of nodes to order. The dashed line shows a curve where the optimal decision is not to order any nodes. The solid line shows a minimum at 477, so the decision is to order nodes.