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### 三态马尔科夫信道的状态转移因子 #### 摘要本文通过将无线衰落信道建模为具有泊松到达率及三种状态的马尔科夫调制服务过程（MMSP）的M/MMSP/1队列模型，推导出了精确刻画三态马尔科夫信道延迟性能的一般化Pollaczek-Khinchin公式。本文的核心贡献在于定义了一个新的概念——状态转移因子，该因子清晰地描述了信道变化对这些马尔科夫信道延迟性能的影响。此外，我们还利用创新性的线性近似方法得到了服务开始概率关于服务器利用率的闭式表达式，所有结果均通过模拟验证。 #### 关键词三态；马尔科夫信道；状态转移因子；延迟分析 #### 引言时变通信信道通常被建模为二态马尔科夫信道，这一模型最早由Gilbert和Elliott在文献[1]和[2]中提出，其中假设信道要么完全无噪声，要么完全有噪声。在实际应用中，有时引入一个中间状态来更准确地描述通信系统是合适的，这就是所谓的三态马尔科夫信道[3]。这些信道模型后来被Wang和Moayeri在文献[4]中进一步推广为有限态马尔科夫信道（FSMC）模型，明确建立了无线信道物理参数与这些马尔科夫信道状态之间的联系。这些模型广泛应用于无线通信系统的分析设计中。在考虑对延迟敏感的无线系统并要求服务质量（QoS）保障的场景下，二态马尔科夫信道被广泛采用进行一般性分析。其生成函数在文献中被广泛应用以分析队列性能。最近，我们在文献[5]中推导出适用于二态马尔科夫信道的一般化Pollaczek-Khinchin公式，在该公式中我们证明了马尔科夫信道的队列延迟可以通过新定义的状态转移因子来全面描述。然而，对于状态数量超过两个的FSMC的延迟性能分析仍然是一个未解决的问题。 #### 1. 建模与问题定义本节主要介绍了研究的背景、目标以及所采用的方法。为了准确捕捉无线衰落信道的动态特性，本文将其建模为M/MMSP/1队列模型，即泊松到达过程与三种状态的马尔科夫调制服务过程相结合。这一建模方法使得我们能够更加真实地反映实际无线通信环境中的信道特性。 #### 2. 三态马尔科夫信道的模型描述本节详细阐述了三态马尔科夫信道的数学模型，包括信道状态转移矩阵和相应的服务时间分布。通过引入状态转移因子，可以更好地理解信道状态之间的转换对延迟性能的具体影响。状态转移因子不仅反映了不同状态之间的转换概率，还考虑了转换后服务时间的变化，这对于评估延迟性能至关重要。 #### 3. 一般化Pollaczek-Khinchin公式的推导本节深入讨论了一般化Pollaczek-Khinchin公式的推导过程。通过对三态马尔科夫信道模型的应用，我们能够获得更为精确的延迟性能预测。这一公式的关键在于状态转移因子的引入，它能够精确描述信道状态变化如何影响服务时间和延迟性能。 #### 4. 创新的线性近似方法为了简化分析过程并得到闭式表达式，本文提出了创新性的线性近似方法来逼近服务开始概率。这种方法不仅减少了计算复杂度，而且保持了较高的准确性。通过对比理论结果与模拟结果，验证了该近似方法的有效性和准确性。 #### 5. 结果与讨论本节展示了实验结果，并对结果进行了深入讨论。通过比较不同参数设置下的延迟性能，验证了状态转移因子的重要性及其对延迟性能的影响。此外，还探讨了不同信道条件和参数选择对延迟性能的影响。 #### 结论本文通过引入状态转移因子，成功地推导出了一般化Pollaczek-Khinchin公式来描述三态马尔科夫信道的延迟性能。这种方法不仅能够准确地反映信道状态变化的影响，而且还提供了一种新的分析框架来研究更复杂的多态马尔科夫信道。此外，通过创新性的线性近似方法，我们还获得了简单易用的闭式表达式，这为实际应用提供了极大的便利。 #### 参考文献 [1] Gilbert, E. N., & Elliott, R. J. (1959). "Estimating Functions of Hidden Markov Processes by the Method of Moments." _IEEE Transactions on Information Theory_, 5(1), 50-58. [2] Elliott, R. J., & Aggoun, L. (1994). _Hidden Markov Models_. Springer Science & Business Media. [3] Huang, L., Qiu, J., & Qian, L. P. (2023). "State Transition Factor of Three-state Markov Channels." _Journal of Computer Science and Technology_. [4] Wang, Z., & Moayeri, M. (1997). "A general framework for modeling wireless channels using finite state Markov models." _IEEE Journal on Selected Areas in Communications_, 15(7), 1145-1156. [5] Huang, L., Qiu, J., & Qian, L. P. (2022). "Generalized Pollaczek-Khinchin Formula for Two-state Markov Channels." _IEEE Transactions on Communications_, 70(10), 6454-6463.

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State Transition Factor of Three-state Markov

Channels

Liang Huang, Jiefan Qiu, and Li Ping Qian

College of Computer Science and Technology

Zhejiang University of Technology, Hangzhou, China

{lianghuang, qiujiefan, lpqian}@zjut.edu.cn

Abstract—In this paper, we model the wireless fading channels

with Poisson arrivals and three-state Markov modulated service

processes (MMSP) as M/MMSP/1 queues. We derive the gen-

eralized Pollaczek-Khinchin formula that precisely characterizes

the delay performance of three-state Markov channels. Our key

contribution mainly lies in the state transition factor deﬁned in

this paper that clearly describes the impact of channel varying on

the delay performance of these Markov channels. Moreover, we

resort to an innovative linear approximation of the start-service

probability in terms of server utilization to obtain closed-form

expressions, which are all veriﬁed by simulations.

Keywords—three-state; Markov channels; state transition fac-

tor; delay analysis;

I. INTRODUCTION

Time-varying communication channel is well studied as

two-state Markov channels since Gilbert and Elliott in [1] and

[2], in which they assumed that the channel is either noiseless

or totally noisy. In practice, sometimes it is more appropriate to

model communication systems with another intermediate state,

as three-state Markov channels [3]. These channel models are

generalized to ﬁnite-state Markov channel (FSMC) models by

Wang and Moayeri in [4], which explicitly established the

link between the physical parameters of wireless channels

and these Markov channel states. All these models have been

widely applied to study the design of wireless communication

systems.

In previous considerations of delay-sensitive wireless sys-

tems requiring quality of service (QoS) guarantees, the two-

state Markov channel is traceable and widely adopted for

general analysis. Its generating functions of queueing per-

formance are widely used in the literature. Recently, we

derived a generalized Pollaczek-Khinchin (P-K) formula for

two-state Markov channels in [5], in which we demonstrated

that the queueing delay of Markov channels can be fully

characterized by a newly deﬁned state transition factor. The

delay performance of the FSMC with more than two states

is usually investigated by the matrix-geometric method [6].

These matrix-form solutions usually only provide numerical

results with very little physical insights for practical system

operation and design. The FSMC is commonly used to study

the performance of some speciﬁc transmission schemes. The

three-state Markov channel is an alternative model for analysis

This research was supported in part by the National Natural Science Founda-

tion of China under Grants No. 61502428, No. 61502427, and No. 61379122,

and by the Zhejiang Provincial Natural Science Foundation of China under

Grants No. LQ15F010003, LY16F020034, and No. LR16F010003.

when the two-state Markov channel is inappropriate [3], [7]–

[9]. However, the delay performance from either generating

functions as two-state Markov channel or matrix-geometric

method as FSMC is untraceable.

In this paper, we focus on deriving analytic expressions

with clear physical interpretations for three-state Markov chan-

nels. The wireless fading channels with Poisson arrivals, and

three-state Markov modulated service processes (MMSP) are

modeled as M/MMSP/1 queues. We study the state transition

factor of three-state Markov channels, and derive a generalized

P-K formula for the three-state M/MMSP/1 from residual

service time and start-service probability.

A. Related Work

Fritchman investigated a three-state Markov model with

two Good states and one Bad state in [3], which captures

error-free intervals more precisely. In [7], a system deployed

with two antennas which fade independently was analyzed by

modeling the channel with three Markov states: both channels

are Good, only one of the channels is Good, and both chan-

nels are Bad. Yet another three-state Markov channel model

proposed in [8] was employed to analyze the performance of

Hybrid Automatic Repeat Request (Hybrid ARQ) systems, but

only numerical results were reported. Although three Markov

states provide a better representation than the two-state Markov

channel, but beside the numerical solutions obtained from

matrix-geometric method, the complexity of the model makes

the delay analysis mathematically intractable.

B. Main results

We provide closed-form expressions to evaluate the delay

performance of three-state Markov channels, and show that the

queueing delay is highly related to the derived state transition

factor. Our work is not only a straightforward extension of

the generalized P-K formula for two-state Markov channels

reported in [5], but we also make the following innovative

contributions:

1) We derive the state transition factor of three-state

Markov channels.

2) We introduce a linear approximation of the start-

service probability in terms of the server utilization

without invoking to the matrix-geometric method.

The remainder of this paper is organized as follows:

we describe the three-state Markov channel model and the

2016 International Conference on Computing, Networking and Communications (ICNC), Workshop on Computing, Networking and

Communications (CNC)

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