Stability of size-based scheduling in resource-sharing networks

1. Stability of size-based scheduling in resource-sharing networks Maaike Verloop CWI & Utrecht U. Sem Borst CWI & Eindhoven U.T. & Lucent Bell Labs Sindo Núñez-Queija CWI & Eindhoven U.T

2. server queue users Introduction • Size-based scheduling in single resource systems • SRPT, LAS, … • Data flows: simultaneous resource possession • Not work conserving • Performance [Yang & De Veciana] • Performance measures • Stability • Delay • Resource occupancy • Compare re-entrant lines and interacting dynamical systems

3. 0 1 2 Introduction • Size-based scheduling in single resource systems • SRPT, LAS, … • Data flows: simultaneous resource possession • Not work conserving • Performance [Yang & De Veciana] • Performance measures • Stability (not trivial) • Delay • Resource occupancy • Compare re-entrant lines and interacting dynamical systems

4. Outline • Model description • Stability of size-based scheduling • SERPT: Shortest Expected Remaining Processing Time • SRPT: Shortest Remaining Processing Time • LAS: Least Attained Service

5. class 0 class L class 1 class 2 class 3 Model description • Linear network • L nodes, with capacity 1 • L+1 classes of users • Poisson arrival processes with rate λi • Random flow size Bi with mean βi • Traffic load ρi= λiβi • Ni denotes the number of class-i flows in the system

6. 0 1 2 3 L Stability • Class i is stable iff P(Ni=0) > 0 • Network is stable if all classes are stable • Necessary condition forstability of network: ρ0+ρi< 1 for all i • Sufficient condition (no parallelism): ρ0+ρ1+…+ ρL < 1 for all i

7. standard conditions 0 1 2 3 L Stability conditions depend on disciplines • Prioritize class 0 • Class i is served only if class 0 is empty • Stable iff ρ0+ρi<1, for all nodes

8. standard conditions 0 1 2 3 L Stability conditions depend on disciplines • Prioritize class 0 • Class i is served only if class 0 is empty • Stable iff ρ0+ρi<1, for all nodes • Prioritize all classes 1,…,L • Class 0 is served only if classes 1,…,L are empty • Stable iff • More stringent stability condition

9. 0 1 2 3 L Size-based scheduling I: SRPT • Class 0 is served at full rate if a class-0 user has the shortest remaining size among all users • Otherwise, at each node i, class i is served at full rate • If Ni > 0, node i works at full capacity, • Class i is stable iff ρ0+ρi < 1 • Stability condition for class 0 • Largest flows that get through • ρ0 (x0) + ρi(xi ) ≤1 • x0≤ xi

10. SRPT: Stability of class 0 • Time-scale decomposition: large class-0 flows • Arrival rate: λ0(ε)= ελ0 • Service requirements: B0(ε)=B0/ε • Traffic load independent of ε: ρ0(ε)= ελ0β0/ε =ρ0 • Distinguish between class-i flows that are larger or smaller than 1/√ε • Calculate P(no i-flow is smaller than 1/√ε) Class 0 is stable in the ε-system for εsmall enough

11. SRPT: Stability of class 0 (cont.) • Short class-0 flows • Assume that class-0 flows are shorter than those of all other classes: M0 < mi (almost strict prioritization) Then class 0 is stable under standard conditions: ρ0+ρi<1

12. 0 1 2 3 L Size-based scheduling II: LAS • In each node a flow has the right to a share of the capacity if it is one of the shortest • Class-0 flows can only utilize the smallest share along the route • Surplus capacity is re-allocated to the other classes  if Ni > 0, node i works at full capacity Class i is stable iff ρ0+ρi < 1

13. LAS: Stability of class 0 ε-system: relatively large class-0 users • Arrival rate: λ0(ε)= ελ0 • Service requirements: B0(ε)=B0/ε • Load independent of ε: ρ0(ε)= ρ0 • Distinguish between “long” and “short” flows Class 0 is stable in the ε-system for εsmall enough

14. N2=0 N1 class 1 class 0 N0 Conclusion • Size-based schedulers may render poor performance in networks • Study performance of schemes such as α-fair allocations that are known to ensure stability • Optimal allocation schemes needed to provide a sensible benchmark • Complexity / approximations • Linear network • More general networks

15. http://www.cwi.nl/~sindoStability of size-based scheduling in resource-sharing networks Maaike Verloop Sem Borst Sindo Núñez-Queija