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We study the problem of detecting a target that moves between a hiding area and an operating area over multiple fixed routes. The research is carried out with one or more cookie-cutter sensors with stochastic intermission, which turn on and off stochastically governed by an on-rate and an off-rate. A cookie-cutter sensor, when it is on, can detect the target instantly once the target comes within the detection radius of the sensor. In the hiding area, the target is shielded from being detected. The residence times of the target, respectively, in the hiding area and in the operating area, are exponentially distributed and are governed by rates of transitions between the two areas. On each travel between the two areas and in each travel direction, the target selects a route randomly according to a probability distribution. Previously, we analyzed the simple case where the sensors have no intermission (i.e., they stay on all the time). In the current study, the sensors are stochastically intermittent and are synchronized (i.e., they turn on or off simultaneously). This happens when all sensors are affected by the same environmental factors. We derive 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 traveling between the two areas. Based on the mean time to detection, we evaluate the performance of placing the sensor(s) to monitor various travel route(s) or to scan the operating area.

Search and detection theory has a history of principal importance in operations research. It has fundamental military and civilian applications such as anti-submarine warfare, counter-mine warfare, and search and rescue operations [

Nowadays when combating piracy at sea, detecting and intercepting threat objects (boats) such as terrorists and drug or weapon smugglers, or securing coastlines and trade routes, it is important to understand the behavior of a target and plan a search and detection strategy accordingly. In a recent work [

In this paper we would like to extend our earlier study to include stochastically intermittent sensors in detecting a target moving between a hiding area and an operating area.

We consider the search problem in which a target moves between a hiding area and an operating area via constrained pathways, as depicted in

In the search problem, variable number of synchronized intermittent cookie-cutter sensors are used to detect the target. We will first introduce the mathematical model for the simple case of non-intermittent sensors (i.e., they stay on all the time) and then extend the model to accommodate the stochastic intermission of the sensors.

We start by specifying the target behavior. The target moves stochastically between the hiding area and the operating area according to the following rules.

・ The dwell time of the target in the hiding area is exponentially distributed with rate

・ On its travel from the hiding area to the operating area, the target takes route k with probability

・ The dwell time of the target in the operating area is exponentially distributed with rate

・ On its travel from the operating area back to the hiding area, the target chooses route k with probability

・ The travel time between the operating area and the hiding area is negligible in comparison with the dwell times in the hiding area and the operating area. Mathematically, we treat the travel time along a route as zero.

The target’s travel between the two areas is mathematically described by a Markov process of two states, with

forward rate

Next, we describe the interaction between the target and sensors. A non-intermittent cookie-cutter sensor is an ideal sensor which detects the target instantly once the target comes within distance R to the center of the sensor where the radius R is called the detection radius of the sensor. When the target is outside the detection radius, it is not detected. In this study, we assume that the detection radius of sensors is large enough to cover the full width of any one of the given routes. Consequently, if a non-intermittent sensor is assigned to monitor a route and the target happens to move along that route, the target will definitely be detected by the sensor. Of course, the situation will be different for an intermittent sensor that turns on and off stochastically.

When a sensor is used to search the operating area, when the sensor is on, and when the target is in the operating area, the interaction between the target and the sensor is modeled using a detection rate

This detection rate is affected by the size of the operating area, and by the detection radius and the speed of the sensor.

When one or more non-intermittent sensors are deployed to monitor one or more routes or to search the operating area, the transitions and detection of the target are governed by a 3-state Markov process of the same parameter form as the one shown in

・ When only one sensor is deployed and it is placed to monitor route k, this gives

・ When only one sensor is deployed and it is used to search the operating area, this corresponds to

・ When two sensors are placed to monitor, respectively, routes k and j (

・ When two sensors are both used to search the operating area, we find

・ When one sensor is placed to monitor route k and a second sensor is used to search the operating area, this gives

To facilitate our analysis, we divide all transition rates by the sum

The parameter form of the normalized Markov process is shown in

In the absence of sensors, at equilibrium, the probabilities of the target being in the hiding area or the operating area are, respectively,

Let T denote the time to detection (random variable), and

We derive two equations for

Using the law of total expectation, we have

which, when divided by

This is an equation for

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

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

As in the previous section, we divide all rates by

where

Note that the parameter

That is, we will focus on the case of

to equilibrium between the on- and off-states;

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

・ 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.

Again, let T represent the time to detection (random variable), and

We derive four equations for

The law of total expectation gives us

Dividing both sides by

This is an equation for

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

We derive asymptotic solutions for the mean time to detection for small

Substituting this asymptotic form into Equations (12)-(15) and examining terms of the order

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

and

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

System (24) is of the same parameter form as system (5) except that p, q, and

We introduce quantity

Next, we calculate coefficients

Substituting the asymptotic form (16) into Equations (22)-(23), collecting all terms of the order

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

The normalized transition rates and parameters (shown in

We examine behaviors of the mean time to detection,

Observation 1: t^{(0)} is a decreasing function of m.

We differentiate

In the derivative above, the right-hand side is negative because factors

Observation 2: t^{(0)} is a decreasing function of p and a decreasing function of q.

We differentiate

Both

This property of

Observation 3: While (p + q) keeps fixed,

In the expression of

When the product

Observation 4: t^{(0)} is a decreasing function of b.

We differentiate

This property of

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

which is a decreasing function of

Observation 6: In general, t^{(0)} is not necessarily a decreasing function of a.

When the condition

Next we study how the mean time to detection changes with

Observation 7: t^{(1)} is always positive.

We write

where h is a function of

To prove that

As a first step, we establish that h is an increasing function of

Thus, to prove

In the above, we have used the facts

Therefore, we conclude that

Finally, we demonstrate the accuracy of the asymptotic solution for the mean time to detection.

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.

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.

HongyunWang,HongZhou, (2016) Performance of Stochastically Intermittent Sensors in Detecting a Target Traveling between Two Areas. American Journal of Operations Research,06,199-212. doi: 10.4236/ajor.2016.62021