Stability of size-based scheduling in resource-sharing networks

Maaike VerloopCWI & Utrecht U.

Sem Borst

CWI & Eindhoven U.T. & Lucent Bell Labs

Sindo Núñez-Queija

CWI & Eindhoven U.T

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– Delay– Resource occupancy

• Compare re-entrant lines and interacting dynamical systems

server

users

queue

3

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

1 2

0

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

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

class 0

class 2 class 3class 1 class L

6

Stability

• Class i is stable iff P(Ni=0) > 0

• Network is stable if all classes are stable

• Necessary condition for stability of network:

ρ0+ρi< 1 for all i

• Sufficient condition (no parallelism):

ρ0+ρ1+…+ ρL < 1 for all i

1 2 3 L

0

7

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

1 2 3 L

0

standard conditions

8

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

L

iiLNN

110 10,,0P

1 2 3 L

0

standard conditions

9

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

1 2 3 L

0

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 ε small enough

L

ii

10 1

11

• 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

SRPT: Stability of class 0 (cont.)

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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

1 2 3 L

0

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 ε small enough

L

ii

10 1

14

class 0

class 1N1

N0

N2=0

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

http://www.cwi.nl/~sindo

Stability of size-based scheduling in resource-sharing networks

Maaike Verloop

Sem Borst

Sindo Núñez-Queija