Stochastics and StatisticsPerformance analysis of a GI-Geo-1 buffer with a preemptive resume priority scheduling discipline
Introduction
In recent years, there has been much interest devoted to incorporating multimedia applications in IP networks. Different types of traffic need different quality of service (QoS) standards, but share the same network resources, such as buffers and bandwidth. For real-time applications, it is important that mean delay and delay-jitter are bounded, while for non-real-time applications, the loss ratio (LR) is the restrictive quantity.
In general, one can distinguish two priority strategies, which will be referred to as Delay priority and Loss priority. Delay priority schemes attempt to guarantee acceptable delay boundaries to delay-sensitive traffic (such as voice/video). This can be achieved by giving it head-of-line (HOL) priority over non-delay-sensitive traffic, and/or by sharing access to the server among the various traffic classes in such a way so that each can meet its own specific delay requirements. Several types of Delay priority (or scheduling) schemes (such as weighted-round-robin (WRR), weighted-fair-queueing (WFQ)) have been proposed and analyzed, each with their own specific algorithmic and computational complexity (see e.g. [11], [14] and the references therein). On the other hand, Loss priority schemes attempt to minimize the packet loss of loss-sensitive traffic (such as data). An overview and classification of some Loss priority (or discarding) strategies can be found in [3], [11].
In this paper, we will focus on the effect of a specific type of Delay priority schemes, i.e., we will analyze a queueing system with a preemptive resume priority scheduling discipline. We assume that delay-sensitive traffic has preemptive priority over delay-insensitive traffic, i.e., when the server of the queueing system becomes empty, a packet of delay-sensitive traffic, when available, will always be scheduled next. In the remaining, we will refer to the delay-sensitive and delay-insensitive traffic as high and low priority traffic respectively. Newly arriving high priority traffic interrupts the transmission of a low priority packet that has already commenced, and the interrupted low priority packet can resume its transmission when all the high priority traffic has left the system, i.e., the part of the packet that was already transmitted before the interruption by high priority packets, does not have to be retransmitted.
In the literature, there have been a number of contributions with respect to queues with a priority scheduling discipline. An overview of some basic priority queueing models can be found in [7], [13], [16], [17], and the references therein. Khamisy and Sidi [8], Laevens and Bruneel [10], Takine et al. [19] and Walraevens et al. [20] have studied discrete-time priority queues with deterministic service times equal to one slot. Khamisy and Sidi [8] analyses the system contents of the different classes, for a queue fed by a two-state Markov modulated arrival process. Laevens and Bruneel [10] analyses the system contents and cell delay in the case of a multiserver queue. In Takine [19], the system contents and the delay for Markov modulated high priority arrivals and geometrically distributed low priority arrivals are presented. Walraevens et al. [20] studies the system contents and cell delay, in the special case of an output queueing switch with Bernoulli arrivals. All these models have a packet transmission time of a single slot in common. Furthermore, preemptive resume priority queues have been analysed in Machihara [12], Sandhu and Posner [15] and Takine and Hasegawa [18]. Machihara [12] analyzes waiting times when high priority arrivals are distributed according to a MAP process. Sandhu and Posner [15] analyses a preemptive resume priority system where the high priority packets cannot be stored in the queue. Takine and Hasegawa [18] studies the waiting times of customers arriving to a queue according to independent MAP processes.
In this paper, we analyze the system contents and packet delay of high and low priority traffic in a discrete-time single-server buffer with a preemptive resume priority scheduling discipline and per-slot i.i.d. arrivals. The transmission times of the packets generated by both types are assumed to be geometrically distributed (with class-dependent parameters). This is mainly done to make the analysis tractable, since the geometrical distribution has the well-known memoryless property. A negative side-effect is of course that, if transmission times of some types of customers are not geometric or arrivals are not i.i.d., the analysis presented in this paper can merely be used as an approximation. However, even then some important approximate performance measures that are practically useful can be calculated, and some important qualitative conclusions can be drawn from the analysis of this model.
As far as the model is concerned, the main difference with the articles involved with HOL priority queues listed above is that the arrival processes of the different types of packets are not mutually independent. This type of arrival process occurs for instance in a multiclass output-queueing router/switch. Therefore the different classes cannot be analyzed separately (i.e., as a model with server interruptions for low priority cells, see e.g. [5]), which complicates the analysis.
We will furthermore demonstrate that an analysis based on generating functions is extremely suitable for modelling this type of buffers with a priority scheduling discipline. From these generating functions, expressions for some interesting performance measures––such as means, variances and approximate tail probabilities of system contents and packet delay––can be calculated. Determining the tail behavior of the system contents and packet delay is one of the main contributions of the paper. Although these are important quantities in the evaluation of the QoS of high and low priority packet streams, this has received only little attention up till now. We will also show that the distribution of the system contents and packet delay of low priority packets not necessarily has a geometric asymptotic behavior.
The remainder of this paper is structured as follows. In the following section, we present the mathematical model. In 3 System contents, 4 Delay, we will then analyze the steady-state system contents and packet delay of both classes. In 5 Calculation of moments, 6 Tail behavior, we calculate expressions for some moments and approximate tail probabilities respectively of the system contents and packet delay of both classes. Some numerical examples are treated in Section 7. Finally, some conclusions are formulated in Section 8.
Section snippets
Mathematical model
We consider a discrete-time single-server system with infinite bufferspace. Time is assumed to be slotted. There are two types of packets arriving to the system, namely packets of class 1 and packets of class 2. The number of arrivals of class j during slot k are independent and identically distributed (i.i.d.) and are denoted by aj,k (j=1,2). Their joint probability mass distribution is defined asNote that the number of arrivals of both classes during one slot can be
System contents
We denote the system contents of class j packets at the beginning of slot k by uj,k (j=1,2), i.e., at the beginning of slot k there are uj,k class j packets in the system.
Their joint pgf is defined asClearly the set {u1,k,u2,k} forms a Markov chain, since the arrival process is i.i.d. and the service times are geometrically distributed. The following system equations can be established:
- 1.
If u1,k=u2,k=0:i.e., the only packets present in the system
Delay
The packet delay is defined as the total amount of time a packet spends in the system, i.e., the number of slots between the end of the packet's arrival slot and the end of its departure slot. We can analyze the packet delay of class 1 packets as if they are the only packets in the system. This is e.g. done in [1] and the pgf of the packet delay of class 1 packets is given by
The analysis of the packet delay of a class 2 packet is more complicated. The
Calculation of moments
The functions Y(z), V1(z) and V2(z) can only be explicitly found in case of some simple arrival processes. Their derivatives, necessary to calculate the moments of the system contents and the packet delay, on the contrary, can be calculated in closed-form. For example, Y′(1) is given by Eq. (4) and the first derivatives of Vj(z) for z=1 are given bywith j=1,2. Let us define λij aswith i,j=1,2. Now we can calculate the mean total system contents, the
Tail behavior
Not only the moments of the system contents and packet delay are important performance measures, but also, and especially, the tail distribution of these quantities, which are often used to impose statistical bounds on the guaranteed QoS for both classes.
From the generating functions of the total system contents, and of the system contents and packet delay of class 1 and class 2 packets derived in 3 System contents, 4 Delay, approximations of the tail probabilities can be derived using complex
An N×N switch
In this subsection, we present some numerical examples. We assume the traffic of the two classes to be arriving according to a two-dimensional binomial process. Its two-dimensional pgf is given byThe arrival rate of class j traffic is thus given by λj (j=1,2). This arrival process occurs for instance at an output queue of an N×N output queueing switch fed by a Bernoulli process at the inlets (see [20]). Notice also that if N→∞, the arrival process is a
Conclusion
In this paper, we have analyzed a discrete-time GI-Geo-1 queue with a preemptive resume priority scheduling discipline and two priority classes. We have derived the joint generating function of the system contents of both classes and the generating functions of the packet delay of both classes. These pgfs are not explicitly found, but we have shown that the moments and the (approximate) tail probabilities of the system contents and packet delay can be found explicitly in terms of the system
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