Mathematical Problems in Engineering

Volume 2016, Article ID 5915918, 21 pages

http://dx.doi.org/10.1155/2016/5915918

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

^{1}Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China^{2}Chongqing 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 [14–16]. 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.