A discrete-time queueing model of EMD policy in high-speed networks

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Abstract

In this paper, we consider a discrete-time queue for the Early Message Discarding (EMD) policy in high-speed networks under bursty arrival. This system is analyzed as a quasi-birth-and-death (QBD) process. We derive the steady-state probability distribution of buffer content by using generating-functions approach. Finally, we obtain the useful performance measure—goodput ratio and give some numerical results.

Introduction

In high speed network, often a set of consecutive packets are grouped into a message (or frame) and loss of one packet results in the loss of the whole message. For example, in ATM where a transport layer protocol (known as AAL) is responsible for this grouping. Many of the selective message discarding policies as means for congestion avoidance were proposed in [1], [2]. Two main policies are Partial Message Discarding (PMD) and Early Message Discarding (EMD). In PMD, the buffer discards packets that belong to message that were already damaged. That is to say, if the buffer is full when a packet arrives, this packet and all consecutive packets that belong to the same message are discarded. The EMD policy is the improvement of PMD policy. In EMD, in addition to the forced discard policy executed when the buffer overflows (as in PMD), there is a threshold (such as K in this paper) represents the buffer occupancy level. If a message begins to arrive when the buffer content is above this threshold, then all packets of this message are discarded. So when K = N, EMD becomes PMD. There have been some papers on the selected packets discarding policy (see [5], [6], [7]). In these contributions, continue-time queueing models were considered: M/M/1/N and On–Off models with PMD and EMD were considered in [5]. Complete solution of M/M/1/N with EMD was given in [6] and fluid analysis of EMD in [7]. Here, we will give the discrete-time queue analysis.

Server interruptions are an abstraction of temporary server unavailability caused by sharing a common server with other queues—e.g., for poling system [3], or multi-class queueing systems—or by external causes, e.g., for maintenance or due to failures or interference [4].

In this paper, we present a discrete-time queueing analysis of the EMD policy with server interruptions and bursty arrival. The outline is as follows: In next section, the mathematical model is described in detail. In Section 3, the probabilities of buffer content under EMD policy are derived. Finally, the useful performance measure—goodput ratio and some numerical results are given in Section 4.

Section snippets

Mathematical description

We consider a discrete-time queueing system, i.e., the time axis is divided into fixed length interval called slots. During each of these slots, packets that arrive in the system are stored in a buffer with finite capacity N, and are served on a FIFO basis. Further, the slots are marked by 0, 1,  , n, … in order. A potential arrival occurs in (n, n+) and a potential departure takes place in (n−, n). More specifically, we consider an early arrival system (EAS). For details, see [9]. Service of a

Steady-state probability distribution

Analyzing the system, we can draw the state transition structure in Fig. 2. Ulteriorly, we obtain the transition probability matrix P as follows:P=A0B1C1A1B10C1A1B1CKAKBK0CKAKBKCNAN,whereA0=γαα¯0γδ¯δ0γαβ¯α¯αβ,B1=γ¯α00γ¯δ¯00γ¯αβ¯00,C1=0γα¯00γδ000γαβ,A1=γαγ¯α¯0γδ¯γ¯δ0γαβ¯α¯γ¯αβ,AK=γαβγ¯α¯γ¯αβ¯γδ¯βγ¯δγ¯β¯δ¯0α¯γ¯α,BK=γ¯αβ00γ¯δ¯β00000,CK=0γα¯γαβ¯0γδγδ¯β¯00γα,AN=0γ¯α¯γ¯α0γ¯δγ¯δ¯0α¯γ¯α,CN=0γα¯γα0γδγδ¯00γα.The stochastic matrix P is the structure matrix and the system is quasi-birth-and-death

Performance measure and numerical results

The basic performance measure for the study of message discarding policies is the effective throughput, which is the ratio of good packets on the outgoing link to the total outgoing flow, such as [2], [10], etc. Lapid [5] firstly introduced the goodput ratio in 1998, which is more suitable performance measure. The goodput ratio is the ratio between packets comprising good messages exiting the system and the total arriving packets at its input denote G. Let M denote the length of an arriving

References (10)

  • M.L. Chaudhry et al.

    Computing waiting-time probabilities in the discrete-time queue: GIX/G/1

    Performance Evaluation

    (2001)
  • S. Floyed et al.

    Random early detection gateways for congestion avoidance

    IEEE/ACM Transactions on Networking

    (1993)
  • S. Floyed, A. Romanow, Dynamics of TCP traffic over ATM networks, in: Proceedings of ACM SIGCOMM, 1994, pp....
  • H. Takagi, A surrey of queueing analysis of polling models, in: Proceedings of the Third IFIP International Conference...
  • T. ALtiok

    Queueing modeling of a single processor with failures

    Performance Evaluation

    (1988)
There are more references available in the full text version of this article.

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The research was supported by the National Natural Science Foundation of China with Grant 10501014.

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