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

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.


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: (1) 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].(2) 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 [14][15][16].Following are the two questions that need our attention: (1) there are many factors that affect DE, such 2 Mathematical Problems in Engineering 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?(2) 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: (1) 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 (worstcase 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.(2) 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.(3) 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.(4) 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.

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].
(1) 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.
(2) 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 are where  = 1/ mean , where  mean 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  = /,  < 1 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].
(3) Suppose that the waiting times of BMs are also exponentially distributed at rate ]; then their probability density function and distribution function are where  = 1/ mean , where  mean is the mean waiting time.
(4) 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: (1) 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 (2) 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. 0 : all target channels are idle; there is no BM in the system.

M/M/N Queueing Models
1 : 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  0 : all target channels are idle; there is no BM in the system: 1 : 1 target channel is busy; there is 1 BM in the system: 2 : 2 target channels are busy; there are 2 BMs in the system: . . . −1 :  − 1 target channels are busy; there are  − 1 BMs in the system: :  target channels are busy; there are  BMs in the 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  0 +  1 +  2 + ⋅ ⋅ ⋅ +   = 1, then the expression for  0 can be written in the form The optimal operation of the queueing system can be analyzed through several performance parameters, some of which follow.

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 3.1.1for  = 1; then, the expression for  0 can be rewritten in the form Performance 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,

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: 0 : all target channels are idle; there is no BM in the system.
1 : 1 target channel is busy; there is 1 BM in the system. .
Then, we have the constraint that ∑ ∞ =0   = 1 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 form When  ≤ , we have When  > , we have

Targets capacity
BMs waiting to be shot The constraint for (24) is With direct substitution of ( 23) and ( 24) into (25), it follows that )) , Then, with substitution of ( 26) into ( 24), we have steady-state probability Performance parameters are as follows: (1) BM loss probabilities ( BMs waiting to be shot): ) . (28) (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 DE 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  = 1.

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  are respectively, where  denotes the total number of types.Let  be the number of layers; the number of target channels deployed along the th layer is denoted by   ; the probability of BMs that will be shot by the -layer defense with short defense depth is

Optimal Allocation of Target Channels
where  is the total number of target channels,  = ∑  =1   .The probabilities of BMs that will be shot by the  layer defense with short defense depth are as follows: 1st layer: ,  1 =  (1)  shoot ⋅   .
(37) 2nd layer: Mth layer: DE of the whole -layer BMDS is Then, we define the optimization problem as finding numbers of layers and target channels deployed along the th 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,  = 5 BMs/min, and  mean = 0.75 min.Tables 1, 2, and 3 are the DE of two-layer, three-layer, and four-layer defense, respectively.
Theorem 1.Let E (M,n (M) ) be the DE of the M-layer BMDS; n (M) = ( 1 ,  2 , . . .,   ) is the allocation plan,   is the number of target channels deployed along the th layer, and  = ∑  =1   ,   ≥ 1.When  is constant, then max E (M,n (M) ) 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  = 5,  = 10 and  = 6,  = 10.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 (+1)th layer; this rule is summarized in Theorem 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 n (M) = ( 1 ,  2 , . . .,   ) ∈ Ω be an allocation plan, and Ω is the feasible region of allocation plans.Suppose that n * (M) = ( 1 ,  2 , . . .,  −1 ,   − 1,  +1 , . . .,   ) ∈ Ω, where the number of target channels deployed along the th layer is Scenario 2. Suppose the SSKP is 0.7, the probability that BMs will be detected by radars is 0.8,  = 5 BMs/min,  mean = 0.75 min, and  * = 65%; 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 n (3) = (5, 3, 2) and n (4) = (4, 3, 2, 1).We can see the least cost is 9 channels for three-layer defense and 8 channels for four-layer defense, and allocation plans are n (3) = (4, 3, 2) and n (4) = (3, 3, 1, 1), 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.
Step 2. Search in the neighborhood of n (M) , (n (M) ), and try to find an allocation plan  1 (n (M) ) satisfying the following: Step 3.
Step 2 and repeat.
Step 5. Output the allocation plan n (M) ; ∑  =1   is the least cost to achieve the demanded DE.

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: (1)  shoot = 1 − (
(52) 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),  (I) =  (II) = 0.7, the probability that BMs will be detected by radars is 0.8,  = 5BMs/min,  mean(I) = 0.75 min, and  mean(II) = 1 min.Table 6 is the DE of three-layer defense.
It can be seen from Table 6 that plan n (3) = (I, I, II) has the biggest DE.We also found that n (4) = (I, I, I, II) is the best, and then we have Theorem 4. Proof.Suppose that we have an allocation plan n I Θ (M) = (I Θ , II Θ ), where  mean(I Θ ) =  mean(II) and  mean(II Θ ) =  mean(I) .
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.  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,  = 5 BMs/min,  mean(I) =  mean(II) = 0.75 min,  (I) = 0.8, and  (II) = 0.6.Table 7 is the DE of three-layer defense.It can be seen from Table 7 that plan n (3) = (I, I, II) has the biggest DE.We also found that n (4) = (I, I, I, II) is the best, and then we have Theorem 6. Mathematical Problems in Engineering   ((+1)+(1− (I) )−(+1)−(1− (II) )) . (69) Obviously, we have The result of Theorem 4 follows.

Numerical Examples
In this section, we use numerical examples to generate some insights into the performance of the proposed queuing models in Section 3. Firstly, using the formula in Section 3.1.2,we draw the relationship of the mean shooting time and intensity of BM arrivals (see Figure 5).Then, we consider using the formula in Section 3.1.1and calculate the loss probability of the M/M/N/N system.We draw the relationship of the BM loss probability and offered density of shootings with  = 1, 2, 3, 4, 5, 6, 8, 12, 16, 20, 30, 40, and 50 (see Figure 6).From Figures 5 and 6, we can see that the probability of BM loss increases with the increasing mean shooting time and intensity of BM arrivals.We set two scenarios: scenario 1 (the number of total BMs is 30, the number of target channels is 3, SSKP is 0.7, the probability that BMs will be detected by radars is 0.8,  = 3 BMs/min, and  mean = 1 min ( = 1)) and scenario 2 (the number of total BMs is 50, the number of target channels is 8, SSKP is 0.8, the probability that BMs will be detected by radars is 0.9,  = 5 BMs/min, and  mean = 1 min ( = 1)).We will firstly simulate the M/M/N/N queueing system and use Matlab to get the figure of the performance of the two scenarios (see Figure 7).Then, we will secondly simulate the M/M/N/C queueing system and use Matlab to get the figure of the performance of the two scenarios (see Figure 8).Table 8 shows the results of operating parameters of the two queueing systems.
In order to explore the changes in the relationship between the performance of systems and different factors, we adjust the parameters in scenario 1, that is, (a) increase the number of arriving BMs, (b) increase the intensity of arriving BMs, (c) increase the mean shooting time for each BM, and (d) reduce the number of target channels.Figure 9 is the queue length as functions of the number of BMs, and Table 9 shows the results of operating parameters of the adjusted queueing system.
Through adjustment of the system configuration, we can observe and summarize the queueing system running condition.This can be useful for decision-making of BMDS operation control and adjusting system configuration.and continued to handle a broader array of queueing scenarios.Several areas for potentially valuable future research have emerged from this work; we suggest the following areas of further research [22][23][24][25][26]. (1) We proposed some approximation in our computations; an important question is the discussion of the accuracy of BM arrivals distribution.
The Poisson arrival process is not the only fitting model for the provided queueing model in our paper.We will consider Bernoulli or Markov BM arrival process in our future research.This also includes relaxing the assumptions of exponential shooting times and allowing waiting times to vary by BM and defense weapon.(2) For the convenience of

Figure 1 :
Figure 1: Long defense depth and short defense depth.

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

Theorem 4 .
Let E (M,n (M) ) be the DE of the M-layer BMDS; n (M) = ( 1 ,  2 , . . .,   ) is the allocation plan,   is the number of target channels deployed along the th layer, and let  1 =  2 = ⋅ ⋅ ⋅ =   .Allocation plan n I (M) = (I, II) indicates that defense weapons of type I are forward-deployed, and n II (M) = (II, I) indicates that defense weapons of type II are forward-deployed.Suppose that defense weapons are identical in the same layer, and  (I) =  (II) and  (I) =  (II) ; if  mean(I) ≤  mean(II) , then one has E (M,n I (M) ) ≥ E (M,n II (M) ) .

Lemma 5 .
Let E (M,n (M) ) be the DE of the M-layer BMDS; n (M) = ( 1 ,  2 , . . .,   ) is the allocation plan,   is the number of target channels deployed along the ℎ layer, and let  1 =  2 = ⋅ ⋅ ⋅ =   .Allocation plan n I (M) = (I, II) indicates that defense weapons of type I are forward-deployed, and n II (M) = (II, I) sh oo tin g tim e In te n si ty o f B M ar ri v a ls

Figure 5 :
Figure 5: Performance parameters as functions of the mean shooting time and the intensity of BM arrivals (M/M/1/1 system).

Target channel number = 1 Figure 6 :
Figure 6: The probability of BM loss as functions of the number of target channels and density of shooting.
BM waiting time and weapons shooting time of scenario 1 and scenario 2 Total number of BMs in queue BM Queue length of scenario 1 and scenario 2Total number of BMs in queue Last BM arrived Maximum capacity of system Total number of BMs in queue and last BM that arrived of scenario 1 and scenario 2

Figure 7 :
Figure 7: Figures of the performance of the M/M/N/N queueing system.
Shooting time for each BM of scenario 1 and scenario 2 Finish time of shooting of scenario 1 and scenario 2

Figure 8 : 20 Mathematical
Figure 8: Figures of the performance of the M/M/N/C queueing system.

Figure 9 :
Figure 9: The queue length as functions of the number of BMs.

Table 1 :
DE of two-layer defense.

Table 3 :
DE of four-layer defense.

Table 4 :
DE of five-layer defense.

Table 6 :
DE of three-layer defense.
indicates that defense weapons of type II are forward-deployed.Suppose that defense weapons are identical in the same layer, and  (I) =  (II) ; if  (I)  mean(I) ≤  (II)  mean(II) , then one has

Table 7 :
DE of three-layer defense.

Table 8 :
Results of operating parameters of the two queueing systems.
M/M/N queueing systems.In addition to the queueing model, four simple rules have been developed for use on the complex channel allocation problems.The main aim of this work is to study a stochastic missile defense process close to the Poisson process and to find allocation rules that

Table 9 :
The results of operating parameters of the adjusted queueing system.