Place reservation: Delay analysis of a novel scheduling mechanism

Abstract

We study the delay performance of a queue with a place reservation mechanism. The objective of this discipline is to provide a better quality of service to arriving packets that are delay sensitive at the cost of allowing higher delays for the best effort packets and was first proposed by Burakowski and Tarasiuk. In our model, we consider a discrete-time queue with arrivals of type 1 (delay sensitive) and type 2 (best-effort). Whenever a packet of type 1 enters the queue, it takes the position of the reservation that was created there by a previous arrival of type 1 and creates a new reservation at the end of the queue. Type 2 arrivals always take place at the end of the queue in the usual way. We present the analysis of this model based on the use of generating functions and provide results for the mean value, variance and tail behaviour of the delay experienced by both the delay-sensitive and the best-effort traffic. For a specific example, we compare the delay performance of this reservation discipline to the performance of an absolute priority discipline on the one hand, and to the reference discipline first-in first-out on the other.

Introduction

A common problem in packet-based communication networks such as Internet protocol (IP) or asynchronous transfer mode (ATM) is the provisioning of adequate quality of service (QoS) guarantees to the traffic flows in the network. Moreover, the specific QoS required by a particular traffic flow often depends on the specific application that generates the flow on a higher layer. For our purposes, we roughly distinguish two kinds of traffic. For real-time applications such as e.g. video or audio streaming, it is important that the end-to-end delay experienced by the data packets is not too large, i.e. the mean delay and delay jitter should be as low as possible. However, these delay-sensitive applications are generally more forgiving towards packets being timed out or lost in the network. Non-real-time applications on the other hand, such as data transfer, demand as little packet loss as possible but can tolerate much larger delays (loss-sensitive applications).

Much effort has been done to equip the nodes in packet-based networks to acknowledge and support this differentiation in QoS requirements [1]. As such, different scheduling algorithms were suggested and implemented in practice, which provide a better QoS to selected flows than the simple first-in first-out (FIFO) scheduling where all packets are regarded as equally important.

A theoretically ideal and fair way to share the server capacity over different input streams is generalised processor sharing (GPS) [2], [3], [4]. However, this mechanism is difficult to apply in packet-based networks, so adaptations for packet scheduling are needed. In the framework of ATM [5] for instance, weighted Round Robin (WRR) and weighted fair queueing (WFQ) [6] were proposed to reduce the delay of certain flows in the node [7]. For these mechanisms, there are separate queues for each type of traffic and the server ‘visits’ each queue in a cyclic and weighted manner. Usually, these scheduling mechanisms require knowledge of the traffic mix to function properly and are difficult to implement. To accommodate delay-sensitive traffic, packet-discarding strategies such as push-out buffer (POB), partial buffer sharing (PBS) [8] and random early detection (RED) [9], [10] have been presented and analysed in literature. Also, in IP-based networks, the introduction and enhancement of the DiffServ [11] and IntServ [12] architecture allows to provide QoS suited to the requirements of specific applications. A simulation study of either is found in [13]. A controllable DiffServ mechanism is proportional differentiated services [14]. Another mechanism that tries to limit the delay of a selected flow is earliest deadline first (EDF) [15], [16] or the use of virtual clocks [17]. An overview can be found in [1], [18].

Suppose we have a queue with two types of packet arrivals and consider the delay experienced by both types of packets. The first arrival flow (type 1) carries delay-sensitive traffic and the second flow (type 2) represents the best-effort traffic or elastic traffic as it is also called. If the queue operates under the FIFO discipline, no special arrangements are made to prioritise the delay-sensitive flow. Therefore, FIFO may serve as the reference discipline for QoS-unaware network nodes. On the other hand, the most extreme way of priority scheduling is absolute priority (AP) or HOL-priority (head of line), either preemptive or non-preemptive [19], [20], [21], [22], [23]. Under this queueing discipline, if the server becomes available and there are type 1 packets present in the queue, a type 1 packet will always be scheduled first, regardless of how many type 2 packets are present and how long they have been waiting for service. This AP discipline was analysed extensively in e.g. [20], from which we draw results for comparison. It is clear that, the server capacity being what it is, AP guarantees the lowest possible delay for the type 1 traffic. However, this comes at the cost of increasing the delay for the packets of type 2. This increase of the delay for the best-effort packets can be very dramatic, especially when the partial load of the prioritised flow is high, and may result in packet starvation or time out.

We study a new and promising delay priority discipline introduced by Burakowski and Tarasiuk [24], that is simple to implement and which reduces the problem of type 2 packet starvation. The idea is to introduce a reserved space (R) in the queue for future arrivals of type 1, as shown in Fig. 1. Whenever a packet of type 1 enters the queue, it takes the position of the reservation that was created there by a previous arrival of type 1 and creates a new reservation at the end of the queue. Type 2 arrivals always take place at the end of the queue in the usual FIFO way. It is seen that this Reservation discipline may allow a 1-packet to jump over some type 2 packets when it is stored in the queue, thus reducing its queueing delay. For instance, with regard to Fig. 1, a new arrival can directly jump to position 3 if it is of type 1, instead of having to queue up at position 7 if it is of type 2. Note that it is impossible for any 1-packets to show up behind the reservation, i.e. to have a position number larger than that of R. As long as it is not seized by a 1-packet, the reserved space R behaves as a normal packet in that it shifts one place to the right every time a packet leaves the server. However, the reserved space cannot enter the server at position 0 nor can it leave the queue. In [25], this idea is carried through further with the proposition of the ‘priority forcing scheme’ (PFS). In this scheme, a certain application may not only send data packets (D-packets) to the queue, but also reservation packets (R-packets). The R-packets are of small size and require very little service time. Their only purpose is to reserve a space in the node for future arriving D-packets. Evidently, the more R-packets are sent by the application in advance, the higher the possible gain in delay performance for the D-flow. Both in [24], [25], the reservation mechanism is studied by means of a simple continuous-time model with Poisson input flows. The analysis in those papers only provides approximated results for the mean delay of both types of packets.

In this paper, we provide a full analysis of the delay of both 1- and 2-packets in a discrete-time queue operating under the Reservation discipline in case the joint arrival process is assumed to be a time-independent process. The analysis is based on a Markovian description of the system state at the beginning of each consecutive slot and is carried out in the z-transform domain, using probability-generating functions (pgfs). We obtain the distribution of the delay experienced by the packets of both types during equilibrium, as well as closed-form expressions for the mean, variance and tail behaviour of those distributions. A first attempt at the analysis of this model can also be found in our previous work [26], [27].

The organisation of the paper is as follows. The mathematical model of a queue with the Reservation discipline is introduced in Section 2, along with some specific assumptions. We identify a sufficient description for the state of the system at an arbitrary slot and provide the main equations that govern its behaviour. In Section 3, the joint pgf of the system state in equilibrium is obtained. Note that some technical proofs are deferred to Appendix A. In Section 4 we derive the exact distribution of the delay distribution of both types of packets, calculate their first- and second-order moment and provide an accurate approximation for their tail distribution. A short numerical example is discussed in Section 5, where we compare the performance of the Reservation discipline with that of the FIFO and AP disciplines. Finally, some conclusions and directions for further work are given in Section 6.