多类OBS节点的阻塞时间分析：一般突发大小分布下的精确计算

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本文主要探讨了多类光学突发交换(Optical Burst Switching, OBS)节点中的阻塞时间问题，这是评估OBS网络性能的关键指标之一。随着电子处理能力和高容量光传输能力的发展，对OBS网络的研究兴趣日益增长。阻塞时间指的是某个通道被某类突发数据流占用的时间长度，它直接影响网络的效率和吞吐量。研究者Shengda Tang、Liansheng Tan 和 Sammy Chan在国际期刊《通信系统》(International Journal of Communication Systems)上发表了一篇论文，他们在2016年通过Wiley Online Library发布，论文的DOI为10.1002/dac.3128。他们采用一种综合的方法，即LC方法（可能是Lightpath Configuration或Load Control），对具有通用突发大小分布的多类OBS节点进行了精确的阻塞概率分析。在理论层面上，作者考虑了一个具有有限视界和单个通道的OBS节点模型，这种模型简化了实际网络的复杂性，但仍然捕捉到了关键的性能特性。他们通过构建数学模型，详细地探讨了不同突发类别、突发大小分布以及节点内部调度策略如何影响阻塞时间的分布和期望值。他们的分析不仅提供了理论上的见解，还进行了数值模拟，以便于工程师们理解和优化网络设计。论文的主要贡献可能包括： 1. **阻塞时间公式**：文中可能给出了多类OBS节点阻塞时间的精确计算公式，基于突发大小分布和节点内部的资源分配策略。 2. **性能边界**：通过分析，论文可能揭示了不同参数组合下，如突发大小、突发类别数量和节点容量，对阻塞时间的极限情况。 3. **算法优化**：如果采用LC方法，可能提出了改进的调度算法或配置策略，以降低阻塞时间，提高网络的整体效率。 4. **适用性研究**：论文可能探讨了阻塞时间分析的通用性和局限性，以及在实际网络中的适用场景。 5. **未来方向**：最后，论文可能会提出对未来研究的建议，比如扩展到多通道模型、动态突发流量环境下的阻塞时间分析等。这篇论文深入研究了多类OBS网络中阻塞时间的重要特性，并通过理论和数值方法提供了一种理解和优化这一关键性能指标的有效工具。对于网络设计者和研究人员来说，这项工作是理解和优化OBS网络性能的重要参考资料。

BLOCKING TIME IN MULTI-CLASS OBS NODES

the blocking time for the horizon-based multi-class OBS nodes and obtain the Laplace–Stieltjes

transforms (LSTs) of the blocking time for horizon-based multi-class OBS nodes, from which the

n-order moments of the blocking time can be derived.

Further, we study the impacts of the offset time and the arrival density on the blocking time, in

which some interesting observations are obtained. For example, an arrival density of a low priority

burst has no inﬂuence on the average blocking time of a higher priority burst. The offset time has

a stronger impact on the lower priority burst than that on the higher priority burst. We believe that

these observations are important in performance evaluation and the design of OBS networks.

We organize the presentation of the current paper as follows. Section 3 presents the analytical

model for the OBS nodes for later analyses. In Section 4, we study the blocking time and obtain its

LSTs. In Section 5, we show a numerical example to discuss the impact of the parameters on the

blocking time and also to verify our theoretical expositions. Finally, Section 6 concludes the whole

paper.

3. MULTILAYER STOCHASTIC FLUID MODEL FOR THE OPTICAL BURST SWITCHING

The current paper will follow the model in [16] but with signiﬁcant extensions. In order to bring the

paper to be self-contained, we ﬁrst give an introduction of the model, and the readers can also refer

to [20]. Throughout the paper, we write A

as the block matrix of A, and let its .i; j /-th element

be ŒA

. We also denote I, 0 and 1 by the unit matrix, zero matrix and the unit column vector with

appropriate dimensions, respectively. We now work on a multi-class OBS node in an OBS network,

which uses the horizon reservation scheme. We assume that the sources generate K classes of bursts

(or BCPs), and we label them from burst-1 to burst-K.Burst-k, 1 6 k 6 K, arrives with an arrival

density 

and follows a Poisson distribution. The Poisson distribution represents the probability of

the number of bursts occurring in a ﬁxed time duration if these bursts occur with a known arrival

density and independently of the time since the last burst. Let X

be the number of burst-k arriving

in time t, then the probability mass function of burst-k is given by

P.X

D m/ D

.



mŠ

This simple yet widely adopted trafﬁc model allows tractable analysis and provides insights for

further study.

Let ı

denote the offset time of burst-k, and we assume the class of the burst with larger identiﬁer

has higher priority. This means that if j<k,thenı

<ı

.LetL

be the length of burst-k,

and it has a phase-type distribution PH.˛

; Q

/: Let q

DQ

1 and denote its transient state

space by

. Without loss of generality, we let the cardinality of M

be n for all k and denote

D¹m

; ;m

º and S

D M

[[M

Let H.t/ be the channel horizon at time t. When some BCP-k (burst-k) arrives at this time epoch

t, this burst-k is blocked immediately if the channel horizon H.t/ > ı

, while it is admitted imme-

diately after its arrival if H.t/ 6 ı

. After the burst-k is admitted, the channel horizon becomes

H.tC/ D H.t/ C L

C ı

, and the channel is occupied for the incoming burst-v (v 6 k) imme-

diately. When BCP-k (burst-k) is blocked, the time duration from the time epoch at which BCP-k

(burst-k) arrives to the time epoch after which the channel is available again for the potential burst-

k is referred to as the blocking time of burst-k [20]. Given the level of horizon is w when burst-k

arrives, we write the blocking time of burst-k as B

.w/.

To illustrate the operation of the multi-class OBS node, a sample path of the channel horizon

process is described in Figure 1(a) as an example when K D 3, the ﬁrst packet (burst-1) arrives at

time epoch t D 0, and it is accepted because the channel is unused and available. When the second

packet (burst-2) arrives at this time epoch, the channel horizon is less than ı

, that is, the channel is

available for transmission of the second packet when it arrives. However, at the time point that the

third packet (burst-1) arrives, the channel horizon is larger than ı

, which means that the channel is

still occupied by the second packet (burst-2) and that the burst-1 is blocked as soon as it arrives. The

other packets can be analyzed similarly.

DOI: 10.1002/dac

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