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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 5915918, 21 pages
http://dx.doi.org/10.1155/2016/5915918
Research Article

Performance Analysis and Optimal Allocation of Layered Defense M/M/N Queueing Systems

1Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China
2Chongqing Nankai Secondary School, Chongqing 400030, China

Received 9 May 2016; Accepted 3 October 2016

Academic Editor: Anna Pandolfi

Copyright © 2016 Longyue Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

One important mission of strategic defense is to develop an integrated layered Ballistic Missile Defense System (BMDS). Motivated by the queueing theory, we presented a work for the representation, modeling, performance simulation, and channels optimal allocation of the layered BMDS M/M/N queueing systems. Firstly, in order to simulate the process of defense and to study the Defense Effectiveness (DE), we modeled and simulated the M/M/N queueing system of layered BMDS. Specifically, we proposed the M/M/N/N and M/M/N/C queueing model for short defense depth and long defense depth, respectively; single target channel and multiple target channels were distinguished in each model. Secondly, we considered the problem of assigning limited target channels to incoming targets, we illustrated how to allocate channels for achieving the best DE, and we also proposed a novel and robust search algorithm for obtaining the minimum channel requirements across a set of neighborhoods. Simultaneously, we presented examples of optimal allocation problems under different constraints. Thirdly, several simulation examples verified the effectiveness of the proposed queueing models. This work may help to understand the rules of queueing process and to provide optimal configuration suggestions for defense decision-making.

1. Introduction

These years, ballistic missile (BM) technology has spread to more and more countries. Nations all over the world are developing missiles capable of reaching enemies. One important mission of strategic defense is to develop an integrated layered BMDS to defend homeland, deployed forces, allies, and friends from ballistic missile attacks [1]. The BMDS is based on a multilayer defense concept and therefore contains more than one defense weapon; it will include different types of defense weapons located on land or ships used to destroy ballistic missiles [2]. Layered BMDS has two advantages: () interception mainly can be divided into 3 phases: boost phase, midphase, and reentry phase. Since the reentry phase is too short and it is the last chance for a shot, BMDS should not rely on a single defense weapon but on defense weapons placed at different locations forming a layered BMDS; the layered BMDS allows for more shot opportunities that will certainly increase the probability of a successful interception [3]. () For given affordable BMs penetration probability (or expected kill probability), cooperation between different missile defense weapons may reduce the expected resources consumption and provide an efficient way of using interceptors. The common methods used in the research on the process simulation and performance evaluation of missile defense are the mathematical programming method [4, 5], the probability calculation method [6], the system simulation method [7], the Markov method [8, 9], and so forth.

Queueing theory is a mathematical theory of stochastic service system which was first proposed by Erlang [10]. Queueing systems have a wide range of applications, such as resource allocation [11], system optimization [12], and communication planning [13]. Similarly, in order to make full use of defense capabilities, queuing theory also has a lot of applications in defense weapons operation research; it can solve problems of weapons configuration or efficiency analysis [1416]. Following are the two questions that need our attention: () there are many factors that affect DE, such as the number of layers, the number of defense weapons, and Single Shot Kill Probability (SSKP); these are also factors that affect the requirement of defense weapons; how are defense weapons, number of layers, BMs, SSKP, and DE interrelated and how can we understand this relationship for achieving the best allocation plan? () If we have deployed different types of defense weapons, then how do we deal with them?

Using M/M/N queueing system to simulate the missile defense process is feasible; the reasons are as follows: () the Poisson process has the simplest mathematical expressions, though BMs arrival is not fully consistent with the Poisson process; it represents the most difficult scenario (worst-case scenario) for the BMDS to deal with. As long as the BMDS can deal with Poisson arrivals, it has certain adaptability to other types of arrivals distribution. () BMs usually have fixed and highly predictable trajectories, though some of them may have limited maneuvering potential; we think this has no influence on our research. () The incoming directions, firing tactics, technical characteristics, and time intervals of BM arrivals have some Poisson features; these can be viewed as customers waiting to be served by servers. () The targets capacity (number of target channels, servers) and shooting times for each target (service times) are limited. When BMs that arrived find that all channels are occupied (not idle) or there was little time for a shot, they will penetrate the BMDS directly. In summary, the M/M/N queueing system can be used to analyze the DE of BMDS, summarize the rules of defense, and provide suggestions for system configuration for defense decision-making. The remainder of the paper is structured as follows. Section 2 proposes the framework for M/M/N queueing model. Section 3 discusses M/M/N queueing models. Section 4 is dedicated to optimal allocation of target channels. Section 5 provides numerical examples. Section 6 includes concluding remarks and future work.

2. M/M/N Queueing Framework

We consider an M/M/N queueing system with BMs arrival rate and shooting rate of defense weapons. M stands for “memoryless” or “Markovian” and means that the process being represented by M comes from an exponential distribution [17].

() Suppose that BMs arrive randomly and independently of each other to a defense weapon and that the average rate at which they arrive is given by the parameter [18]; that is,

This is known as a Poisson arrival process; is the probability that BMs arrive within time . Suppose that the time intervals between arrivals are randomly taken from the exponential distribution with parameter ; their probability density function and distribution function are

The exponential distribution is memoryless, which indicates that the BMs arrivals are random.

() Suppose that the BMs are shot in the order of their arrivals; the shooting time for a BM is also exponentially distributed at rate ; then, its probability density function and distribution function arewhere , where is the mean shooting time. The shooting time depends on the reaction time of the defense weapon and the time of interceptor flying from the launch point to the calculated encounter point, which is related to the technical capabilities of defense weapons. Introducing , means that the queue is stable if the mean shooting time is less than the mean arrival time intervals; it can be understood as the firing density (shooting intensity) [19].

() Suppose that the waiting times of BMs are also exponentially distributed at rate ; then their probability density function and distribution function arewhere , where is the mean waiting time.

() Additionally, if there is an idle target channel when a BM arrives at the system, then the defense weapon will shoot it immediately. In this paper, we divided the queueing system into two types: () loss system (when BMs that arrived find that all target channels were occupied (not idle), they will penetrate the BMDS directly (leave the system without service)) and () mixed system (when BMs that arrived find that all target channels were occupied, BMs will not be penetrated but will wait for a limited time (depending on the time of BMs flying in the killing zone of BMDS) until a target channel becomes available). We use the term “defense depth” to distinguish between the loss system and the mixed system. “Short defense depth” is defined as the case when waiting times of BMs are shorter than shooting times of defense weapons (loss system), and the “long defense depth” is defined as the case when waiting times of BMs are longer than shooting times of defense weapons (mixed system), as shown in Figure 1.

Figure 1: Long defense depth and short defense depth.

3. M/M/N Queueing Models

3.1. Identical Weapons, Short Defense Depth, and M/M/N/N System
3.1.1. Multiple Target Channels and M/M/N/N System

BMDS with short defense depth can be seen as a random loss system that does not wait. Let be the number of target channels of BMDS; that is, the number of BMs shot by defense weapons cannot exceed at the same time; is the detection probability of BMDS radars. Possible states of the system are as follows::all target channels are idle; there is no BM in the system.:1 target channel is busy; there is 1 BM in the system.: k target channels are busy; there are BMs in the system.:all target channels are busy; there are BMs in the system.Figure 2 is the state transition diagram of M/M/N/N system. In Figure 2, the arrow indicates incoming BMs, and the arrival rate is . means target channels are busy, and the state is transferred to when a new BM arrives. Arrow indicates shootings of defense weapons, when the shooting process for a BM is done, and then the state is transferred from to with rate , transferred from to with rate , and transferred from to with rate . Balance equations are as follows::all target channels are idle; there is no BM in the system::1 target channel is busy; there is 1 BM in the system::2 target channels are busy; there are 2 BMs in the system:: target channels are busy; there are BMs in the system:: target channels are busy; there are BMs in the system:

Figure 2: State transition diagram of M/M/N/N system.

Due to the fact that is the probability of the BM found by the radar, we have

is the mean number of BMs found to arrive during the mean shooting time. Because , then the expression for can be written in the form

The optimal operation of the queueing system can be analyzed through several performance parameters, some of which follow.(1)BM loss probabilities (penetration probabilities):(2)Relative probabilities of BMs that will be shot:(3)Absolute probabilities of BMs that will be shot:(4)Mean number of occupied target channels:The mean occupancy rate of target channels is .

3.1.2. Single Target Channel and M/M/1/1 System

The M/M/1/1 queueing system can be viewed as a special case of Section 3.1.1 for ; then, the expression for can be rewritten in the formPerformance parameters are as follows:(1)BM loss probabilities:(2)Relative probabilities of BMs that will be shot:(3)Absolute probabilities of BMs that will be shot:The concept of DE is the product of probabilities of BM that will be shot times SSKP; that is,

3.2. Identical Weapons, Long Defense Depth, and M/M/N/C System

From Section 1, we know that BMDS with long defense depth can be regarded as a stochastic service system with limited waiting time, that is, the mixed queueing system. For each incoming BM, the defense weapons use idle target channels to shoot. When BMs that arrived find that all target channels were occupied, BMs will not be penetrated but will wait for a limited time until a target channel becomes available. Possible states of the system are as follows::all target channels are idle; there is no BM in the system.:1 target channel is busy; there is 1 BM in the system.: k target channels are busy; there are BMs in the system.:all target channels are busy; there are BMs in the system.:all target channels are busy; there is 1 BM waiting.:all target channels are busy; there are BMs waiting.Figure 3 is the state transition diagram of M/M/N/C system. In Figure 3, the arrow indicates incoming BMs, and the arrival rate is . means target channels are busy, and the state is transferred from lower state to higher state when new BMs arrive. Arrow indicates shootings of defense weapons, when the shooting process for a BM is done, and then the state is transferred from higher state to lower state with rate or . Balance equations are

Figure 3: State transition diagram of M/M/N/C system.

Then, we have the constraint that for (21). Let be the mean number of BMs that arrived during the mean shooting time; let be the mean number of penetrated (leaving without being shot) BMs during the mean shooting time. Then, (21) can be written in the formWhen , we haveWhen , we haveThe constraint for (24) isWith direct substitution of (23) and (24) into (25), it follows thatThen, with substitution of (26) into (24), we have steady-state probabilityPerformance parameters are as follows:(1)BM loss probabilities ( BMs waiting to be shot):(2)Relative probabilities of BMs that will be shot:(3)Absolute probabilities of BMs that will be shot:(4)Mean number of occupied target channels:The mean occupancy rate of target channels isDE is the product of probabilities of BM that will be shot times SSKP; that is,Similarly, the M/M/1/C queueing system can be viewed as a special case for .

3.3. Different Defense Weapons

Defense weapons in Sections 3.1 and 3.2 are identical, and the BMDS may deploy different types of defense weapons. For different types of defense weapons, the waiting time of BMs, the detection probability of radars, and SSKP may be different. In order to model the queueing system, we choose one type of defense weapon as the reference and then substitute the reference defense (type ) for defense weapons of type . The “replacement process” is named equivalent replacement method [20]; basic equations are

Subscripts and in (34) are used to distinguish defense weapons of type from defense weapons of type . From the above equations, we can substitute for ; the intensity of BMs killed by defense weapons of type and the intensity of BMs arrivals for defense weapons of type arerespectively, where denotes the total number of types.

4. Optimal Allocation of Target Channels

4.1. Identical Defense Weapons and M/M/N/N System
4.1.1. Defense Effectiveness

Let be the number of layers; the number of target channels deployed along the layer is denoted by ; the probability of BMs that will be shot by the -layer defense with short defense depth is where is the total number of target channels, . The probabilities of BMs that will be shot by the layer defense with short defense depth are as follows:1st layer:2nd layer:Mth layer: DE of the whole -layer BMDS isThen, we define the optimization problem as finding numbers of layers and target channels deployed along the layer so as to maximize DE, subject to a given set of constraints:For the nonlinear optimization problem (41), when problem size is small, we can use algebra, dynamic programming, or enumeration method to solve it. When the size of the problem is very large, an approximate solution can be obtained by using some advanced algorithms, for example, genetic, neural network, and heuristic algorithms. In order to get some potential and useful allocation rules, we analyze a scenario.

Scenario 1. Suppose that the number of target channels is 10, SSKP is 0.7, the probability that BMs will be detected by radars is 0.8, BMs/min, and min.
Tables 1, 2, and 3 are the DE of two-layer, three-layer, and four-layer defense, respectively.
Let be the allocation plan of target channels, where is the number of layers and is the number of target channels deployed along the layer. From the results of Tables 1, 2, and 3, we noticed that plans , , and have the biggest DE. We also found that , and then we have Theorem 1.

Table 1: DE of two-layer defense.
Table 2: DE of three-layer defense.
Table 3: DE of four-layer defense.

Theorem 1. Let be the DE of the M-layer BMDS; is the allocation plan, is the number of target channels deployed along the layer, and , . When is constant, then is stochastically increasing as increases; that is,

Proof of Theorem 1 is similar to Lemma  1 in [8]. Now, we continue to compute DE of , and , . Tables 4 and 5 are the DE of five-layer and six-layer defense, respectively. Another useful rule is that the number of target channels deployed along the th layer should be not less than th layer; this rule is summarized in Theorem 2.

Table 4: DE of five-layer defense.
Table 5: DE of six-layer defense.

Theorem 2. Let be the DE of the M-layer BMDS; is the allocation plan, is the number of target channels deployed along the layer, and , . For allocation plan , the number of target channels deployed along the layer is decreasing as increases; that is, for all , . When is constant, then one has .

Proof. Suppose we have a plan , where , . In order to prove Theorem 2, we only need to prove , which can be simplified by proving , where , . For functionwhen is constant, decreases with the increase of , and when is constant, increases with the increase of [2, 20]. For proof simplified by letting for all defense weapons, then , , , and arerespectively, where , . DE of the two-layer defense areSince the expansion of is , then the approximate representations of , , , and arerespectively. Because , we have In order to prove Theorem 2, we need to proveWhen , Theorem 2 is true. Obviously, when , we have ; then, , and the result of Theorem 2 follows.

4.1.2. Minimum Requirements of Target Channels

The requirements of target channels necessary to achieve a demanded DE can be viewed as a dual problem of (41). Assuming that the DE is held at greater than , we define the optimization problem so as to minimize the requirements of target channels, subject to a given set of constraints: For the nonlinear optimization problem (49), when problem size is small, we can use algebra, dynamic programming, or enumeration method to solve it. When the size of the problem is very large, an approximate solution can be obtained by using some advanced algorithms, for example, genetic and heuristic algorithms. Then, we give the definition of neighborhood [21].

Definition 3. Let be an allocation plan, and is the feasible region of allocation plans. Suppose that , where the number of target channels deployed along the th layer is . Then, one says that is in the neighborhood of ; that is,

Scenario 2. Suppose the SSKP is 0.7, the probability that BMs will be detected by radars is 0.8, BMs/min, min, and ; then the question becomes as follows: “What is the least cost of target channels to achieve a demanded DE?”
Figure 4 is the schematic of search in the neighborhood of and . We can see the least cost is 9 channels for three-layer defense and 8 channels for four-layer defense, and allocation plans are and , respectively. So, we give a simple algorithm in finding the least cost of DWs to achieve a demanded DE; specific search methods are as follows. A feasible initial allocation plan is very important in this algorithm. Theorem 1 proposes the basic rule of finding an initial plan, which greatly simplifies the searching process.

Figure 4: Search in the neighborhood of and .

Step 1. Suppose the number of layers is ; the demanded DE is ; and give an initial allocation plan , where .

Step 2. Search in the neighborhood of , , and try to find an allocation plan satisfying the following:

Step 3. If , then ; go to Step 2 and repeat.

Step 4. If , go to Step 5.

Step 5. Output the allocation plan ; is the least cost to achieve the demanded DE.

4.2. Different Defense Weapons and M/M/N/N System
4.2.1. Different Defense Weapons and Identical SSKPs

In this section, we consider different types of defense weapons. Let be the number of layers; the number of target channels deployed along the th layer will be denoted by , assuming that the total number of types is and defense weapons are identical in the same layer. As in Section 3.3, subscript indicates defense weapons of type ; then probabilities that BMs will be shot by the -layer defense with short defense depth are as follows:1st layer:2nd layer:th layer: DE of the whole M-layer BMDS is

For certain types and numbers of target channels, we define the optimization problem as finding which type of defense weapon should be deployed on each layer so as to maximize the ED, subject to a given set of constraints:In order to get some potential and useful allocation rules, we analyze a scenario.

Scenario 3. Suppose that the number of layers is 3, the number of defense weapon types is 2 (types I and II), , the probability that BMs will be detected by radars is 0.8, BMs/min, min, and min.
Table 6 is the DE of three-layer defense.
It can be seen from Table 6 that plan has the biggest DE. We also found that is the best, and then we have Theorem 4.

Table 6: DE of three-layer defense.

Theorem 4. Let be the DE of the M-layer BMDS; is the allocation plan, is the number of target channels deployed along the layer, and let . Allocation plan indicates that defense weapons of type I are forward-deployed, and indicates that defense weapons of type II are forward-deployed. Suppose that defense weapons are identical in the same layer, and and ; if , then one has .

Proof. Suppose that we have an allocation plan , where and . In order to prove Theorem 4, we only need to prove . For functionwhen is constant, decreases with the increase of , and when is constant, increases with the increase of [2, 20]. Then, , , , and arewhereDE of the defense areSince the expansion of is , then the approximate representations of , , , and areIn order to prove Theorem 4, we need to proveLet and ; then,Since , then , and the result of Theorem 4 follows.

The terms and that appear in the proof of Theorem 4 could be a factor in channels allocation; we extended Theorem 4 to obtain Lemma 5.

Lemma 5. Let be the DE of the M-layer BMDS; is the allocation plan, is the number of target channels deployed along the layer, and let . Allocation plan indicates that defense weapons of type I are forward-deployed, and indicates that defense weapons of type II are forward-deployed. Suppose that defense weapons are identical in the same layer, and ; if , then one has .

4.2.2. Different Defense Weapons and Different SSKPs

In this section, we consider different defense weapons and different SSKPs of layered BMDS. It is obvious that higher SSKP dose improves the DE. How SSKP affects the target channels allocation is a complex problem; however, we can summarize some useful rules; see Scenario 4.

Scenario 4. Suppose that the number of layers is 3, the number of defense weapon types is 2 (types I and II), the probability that BMs will be detected by radars is 0.8, BMs/min, min, , and .
Table 7 is the DE of three-layer defense. It can be seen from Table 7 that plan has the biggest DE. We also found that is the best, and then we have Theorem 6.