The parameter form of the normalized Markov process is shown in Figure 4.
In the absence of sensors, at equilibrium, the probabilities of the target being in the hiding area or the operating area are, respectively, and. This fact will be used later in the calculation of mean time to detection.
3. Mean Time to Detection in the Case of Sensors without Intermission
Figure 4 describes the general model for the case of non-intermittent sensors. It can accommodate arbitrary number of sensors. For mathematical convenience, we label the hiding area as state 1 and the operating area as state 2. We solve for the mean time to detection.
Let T denote the time to detection (random variable), and denote the state of the target at time t. Time 0 is defined as the time when the sensors are deployed. We examine the conditional mean time to detection given that the target is in state j at time 0.
We derive two equations for and based on the Markov process in Figure 4. Starting with, the probability distribution of is
Using the law of total expectation, we have
Figure 4. The normalized Markov process governing the transitions of the target.
which, when divided by and in the limit of going to zero, yields
This is an equation for and. Similarly, another equation can be derived starting with. The two equations for and are
Solving linear system (5), we obtain
Before the deployment of sensors, the equilibrium distribution of the target is
Thus, the overall mean time to detection has the expression
4. Mathematical Formulation for the Case of Synchronized Stochastically Intermittent Sensors
We extend above discussions to consider synchronized intermittent sensors that stochastically turn on or off simultaneously. We model the stochastic evolution of sensors as a 2-state Markov process with an on-rate and an off-rate, as illustrated in Figure 5.
As in the previous section, we divide all rates by to normalize the problem. After the non- dimensionalization scaling, we write the on-rate and the off-rate as
Figure 5. The 2-state Markov process governing the intermittent sensors.
Note that the parameter defined above corresponds to the equilibrium probability of sensors being on, which is also the fraction of time that sensors are on. The quantity is the equilibrium probability of sensors being off. The choice of notation indicates that we will focus on the parameter regime of.
That is, we will focus on the case of. Note that is the rate of sensors relaxing
to equilibrium between the on- and off-states; is the rate of the target relaxing to equilibrium between the hiding area and the operating area. Parameter in (9) measures the ratio of these two relaxation rates.
We now combine the stochastic travel of the target and the stochastic intermission of the sensors into a 5-state Markov process for the target-sensors system as illustrated in Figure 6. The normalized version of this 5-state Markov process where all transition rates are divided by, is shown in Figure 7. In the model, the sensors’ intermission and the target’s travel between the two areas are independent of each other. The target- sensors system can reside in any of 4 states before the detection. For mathematical convenience, we number the 4 states as follows: (Figure 7).
・ State 1: the target is in the hiding area and the sensors are on.
・ State 2: the target is in the operating area and the sensors are on.
・ State 3: the target is in the hiding area and the sensors are off.
・ State 4: the target is in the operating area and the sensors are off.
Figure 7 gives the general model for the target-sensors system with synchronized stochastically intermittent sensors. It can accommodate arbitrary number of sensors. We study the mean time to detection in this 5-state Markov process.
Again, let T represent the time to detection (random variable), and denote the state of the target at time t. We consider the conditional mean time to detection
We derive four equations for based on the 5-state Markov process in Figure 7. Starting with
, the probability distribution of is
Figure 6. The 5-state Markov process governing the target- sensors system.
Figure 7. The normalized 5-state Markov process governing the target-sensors system. In the non-dimensionalization scaling, all transition rates are divided by.
The law of total expectation gives us
Dividing both sides by and taking the limit as going to zero, we arrive at
This is an equation for. Three other equations are derived using the same approach, starting with, respectively, , , and. Putting together, these four equations for are
Analytical solutions to the above linear system is hard to obtain. Instead, in the next section, we use this linear system to calculate asymptotic expansions for and for the overall mean time to detection when is small.
5. Asymptotic Solutions for the Mean Time to Detection in the Case of Synchronized Intermittent Sensors
We derive asymptotic solutions for the mean time to detection for small. We start by writing the conditional mean time to detection in the asymptotic form of
Substituting this asymptotic form into Equations (12)-(15) and examining terms of the order, which
are of the largest magnitude, we obtain
From Equations (12)-(15), we will derive two equations that do not contain any coefficient of the order
. These two equations will then be used to calculate the leading coefficients in the asymptotic form:
and. We rewrite Equations (12)-(15) to separate terms with coefficients from other terms.
The two equations are constructed by
The resulting two equations are
Substituting asymptotic form (16) into the two equations above, keeping only terms of order, and using result (17), we arrive at
System (24) is of the same parameter form as system (5) except that p, q, and in (5) are replaced by, , and in (24). The solution of (24) immediately gives us the leading coefficients:
We introduce quantity, which will be useful in the subsequent calculation. Quantity
has the expression
Next, we calculate coefficients. Substituting the asymptotic form (16) into equation (20)- (21), and using the fact that all leading coefficients are already known, we find expressions for and.
Substituting the asymptotic form (16) into Equations (22)-(23), collecting all terms of the order, and using results (27)-(28), we obtain two equations for and.
The solution of (30) gives us expressions for coefficients.
Before the sensors are assigned to monitor/search routes, the equilibrium distribution of the target-sensors system is
It follows that the overall mean time to detection has the expression
Therefore, the overall mean time to detection has the asymptotic expansion
where the coefficients and are given by
The normalized transition rates and parameters (shown in Figure 7) are related to the physical transition rates (shown in Figure 6) as follows
6. Behaviors of the Mean Time to Capture
We examine behaviors of the mean time to detection, , as a function of parameters, , , p, q, and.
Observation 1: t(0) is a decreasing function of m.
We differentiate as defined in (33) with respect to and find that
In the derivative above, the right-hand side is negative because factors, p, and q are all positive and bounded by 1. This property of is not surprising. Recall that is the normalized detection rate in the operating area. For a larger, it is reasonable to expect that the target will be detected sooner.
Observation 2: t(0) is a decreasing function of p and a decreasing function of q.
We differentiate, respectively, with respect to p and q and obtain that
Both and are negative, which follows directly from the fact that all terms in the numerators are positive. In particular, it is true that
This property of is also expected. Remember that p and q are, respectively, the probability of the target being detected on its way to the operating area and the probability of being detected on its way back to the hiding area. Increasing either p or q while keeping the other unchanged will speed up the detection of the target.
Observation 3: While (p + q) keeps fixed, becomes an increasing function of (pq).
In the expression of, while the sum is fixed, the combination appears only in the denominator.
When the product is increased, the denominator is decreased, and as a result, is increased. This behavior tells us that when evaluating several options with the same value of, the optimal option is the one with the smallest value of.
Observation 4: t(0) is a decreasing function of b.
We differentiate with respect to to get
This property of is expected. is the fraction of time that the sensors are on. Increasing the value of leads to an increase in the probability that the sensors are on at a random time. and thus reduces the time to detection.
Observation 5: When p = q = 0, t(0) is a decreasing function of a.
When sensors are deployed only to search in the operating area and no sensor is used to monitor any of the routes, we have. In this case, has the simple expression
which is a decreasing function of. When only the operating area is searched, the target can only be detected in the operating area; the target cannot be detected on its travels between the two areas. Consequently, the detection probability increases with the fraction of time that the target spends in the operating area, which is given by.
Observation 6: In general, t(0) is not necessarily a decreasing function of a.
When the condition is not satisfied, may not be monotonic with respect to variable. Figure 8 shows a plot of vs while other parameters are fixed at, , , and. initially decreases as is increased from 0. It attains a minimum around. For, appears to be an increasing function of. It is clear that in general, is not monotonic with respect to.
Next we study how the mean time to detection changes with, the normalized time scale of sensors relaxing to equilibrium between the on- and off-states. For that purpose, we examine coefficient in the asymptotic expansion.
Observation 7: t(1) is always positive.
We write as
where h is a function of, given by
To prove that is always positive, we only need to show is always true.
As a first step, we establish that h is an increasing function of, which is reached by differentiating h with respect to.
Figure 8. Plot of vs for, , , and. Note that in general, is not monotonic with respect to.
Thus, to prove, we only need to show. This can be verified directly:
In the above, we have used the facts
Therefore, we conclude that is always positive. It follows that as a function of while other para- meters are fixed, the mean time to detection decreases as is reduced. In other words, when the fraction of time that sensors are on is fixed (i.e., is kept at a constant), and when the normalized time scale of sensors relaxing to equilibrium between the on- and off-states is reduced (i.e., is made smaller), the detection performance of sensors is improved (i.e., the mean time to detection is shorter).
Finally, we demonstrate the accuracy of the asymptotic solution for the mean time to detection. Figure 9 compares an accurate numerical solution and the asymptotic expansion for, in the range of. It is clear that the dependence of on is well captured in the asymptotic solution even at.
We have addressed the performance of stochastically intermittent sensors when used to detect a target that moves between a hiding area and an operating area via multiple routes. We have derived asymptotic expansions for the mean time to detection when the on-rate and off-rate of the sensors are large in comparison with the rates of the target moving between the hiding area and the operating area. Using the mean time to detection, we have evaluated the performance of placing sensor(s) to monitor various travel route(s) or to scan the operating area.
Figure 9. Comparison of an accurate numerical solution of and the asymptotic solution for, , , , and.
Acknowledgements and Disclaimer
Hong Zhou would like to thank Naval Postgraduate School Center for Multi-INT Studies for supporting this work. Special thanks go to Professor Jim Scrofani and Deborah Shifflett. The authors also thank Mr. Ed Waltz and Dr. Will Williamson for their inspirational suggestions. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.