Gradual Delay Differentiation in Priority Scheduling

1. Gradual Delay Differentiation in Priority Scheduling Tom Maertens, Joris Walraevens and Herwig Bruneel Ghent University (UGent) Department of Telecommunications and Information Processing (TELIN) Stochastic Modelling and Analysis of Communication Systems (SMACS) Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium

2. Framework • Modern telecommunication networks are designed to offer a wide variety of services: • information access • e-mail • internet telephony • file sharing • streaming media • … • Different services have extremely diverse Quality-of-Service (QoS) requirements: • real-time services do not tolerate delay • non-real-time services are quite vulnerable to loss and require a large throughput

3. Service differentiation • The traffic that flows through telecommunication devices nowadays can thus more or less be classified into two types • Delay as QoS-measure: delay-sensitive (type-1) traffic versus delay-tolerant(type-2) traffic • To achieve the required delay differentiation between both types of traffic, delay-sensitive traffic is prioritised in scheduling packets for transmission

4. Assumptions • Physical structure of the system • discrete time • infinite storage capacity, divided in two priority queues • one transmission channel • Transmission process • work-conserving • single-slot transmission times • transmissions are synchronised to the slot boundaries

5. Static priority scheduling Priority is always given to delay-sensitive packets: • delay-tolerant packets can only be transmitted when there are no delay-sensitive packets present in the system • priority levels of both types of traffic never change during time time

6. Performance of the static priority scheduling discipline • Low delays for delay-sensitive packets • Possibly excessive delays for delay-tolerant packets, especially when the system is highly loaded and much network traffic is delay-sensitive Gr*!@#%#$ time

7. Dynamic priority scheduling • Priority levels of the two types of traffic can change during time, so delay-tolerant packets can also be transmitted when there are delay-sensitive packets present in the system • Dynamic priority scheduling disciplines aim for a more gradual delay differentiation between both types of traffic • Two categories: • varying priority levels • priority jumps: the priority level of delay-tolerant packets can increase in the course of time

8. Priority jumps • Packets of the low-priority queue can jump to the high-priority queue in the course of time • Jumpeddelay-tolerant packets are treated in the high-priority queue as if they are delay-sensitive packets • Jumps occur at the end of slots time

9. Jumping criteria: to decide if and when delay-tolerant packets jump • A maximum queueing delay in the low-priority queue • A queue-length-threshold for one of the queues • A random jumping probability per time unit • An arrival characteristic of one type of traffic

10. Jumping mechanisms • Merging the high- and low-priority queues in a slot • every slot: Merge-Every-Slot (MES) • with a certain probability: Merge-By-Probability (MBP) • Letting only one packet jump in a slot • always: Jump-Or-Transmit (JOT) • with a certain probability: Jump-By-Probability (JBP) • Jump in a slot also depends on the number of arrivals • delay-sensitive (type-1) packets: Jump-If-Arrivals-of-type-1 (JIA1) • delay-tolerant (type-2) packets: Jump-If-Arrivals-of-type-2 (JIA2)

11. Example: Merge-By-Probability At the end of a slot, the total content of the low-priority queue jumps with probability to the end of the high-priority queue (i.e., both queues are merged with probability) time

12. Arrival process • Two types of packets • type-1: delay-sensitive • type-2: delay-tolerant • Number of arrivals of both types of packets (denoted by a1 and a2 respectively) • are independent and identically distributed (i.i.d.) from slot to slot • can be correlated within one slot

13. Determining the system equations Describe the evolution of the queue contents from slot to slot • uH,k = content of the high-priority queue at the beginning of slot k • uL,k = content of the low-priority queue at the beginning of slot k

14. Transforming to a functional equation • Introducing probability generating functions: • Assuming that the system evolves towards a steady state ! dropping the time index k

15. Solving the functional equation Determining the constant U(0,0) and the functions U(0,z2) en U(z1,z1) leads to the joint probability generating function of the contents of both queues

16. Delay of a type-1 packet • Jumps always occur at the end of a slot ! all type-1 packets that arrive during a slot enter the high-priority queue in front of jumping type-2 packets • Is only determined by the content of the high-priority queue at the moment of arrival

17. Delay of a type-2 packet • Priority scheduling: new type-1 packets can arrive while type-2 packets are waiting in the low-priority queue, and these type-1 packets have priority • Jumping mechanism: type-2 packets can jump to the high-priority queue in the course of time • Combination of these two characteristics of the model makes the analysis not always straightforward!

18. Results • Probability generating functions of • the contents of the two priority queues • the delays of both types of packets (for most models) • Performance measures: • moments via the moment generating property of probability generating functions • approximate tail distributions via the dominant-singularity method applied on probability generating functions → not necessarily exponentially decaying tail probabilities

19. Numerical examples: packet switch • Arrival at an input port • occurs with probability T (= arrival rate) • and is of type 1 with probability  (= fraction of type-1 arrivals in the overall traffic mix) • Uniform and independent routing towards the output ports

20. Mean delays of both types of traffic for T=0.9

21. Mean delays of both types of traffic for T=0.9

22. Conclusions • Priority schemes with priority jumps build upon the simplicity and efficiency of the static priority scheme, but prevents delay-sensitive traffic from starving • Depending on the delay requirements of the different types of traffic, we can introduce and tailor one of the jumping mechanisms • Analysis based on probability generating functions • can overcome mathematical challenges (e.g., the calculation of boundary functions) • is useful for the calculation of different performance measures (such as moments and tail probabilities)