1. Energy-Aware Scheduling on Heterogeneous Processors
Rhonda Righter, IEOR, UC Berkeley
Joint work with
Osman Akgun, IEOR, UC Berkeley
Doug Down, McMaster University

2. Motivation
Server farms and cloud computing networks
Energy consumption and greenhouse gas emission have become major concerns
Servers vary in their speeds and energy consumption
How do we schedule our work to balance energy consumption and congestion?

3. Model
Identical jobs with holding cost rate h
m servers, exponential service times with service rate μi and server usage cost βi
Decision: Assign a job to an available server or leave it idle to minimize total costs
With/without reassignment (can jobs move?)
With/without job arrivals (clearing system)
Two servers/more than two servers

4. Prior Work – The slow server problem
βi ≡ 0 – servers differ only in μi
Weiss and Pinedo (1980), Agrawala et al. (1984), Lin and Kumar (1984), Walrand (1984), Kumar and Walrand (1986), Coffman et al. (1987), Viniotis and Ephremides (1988), Rosberg and Makowski (1990), Stockbridge (1991), Weber (1993), Xu (1994), Koole (1995), Luh and Viniotis (2002), de Vericourt and Zhou (2006)

5. Prior Work – The slow server problem
Clearing system (no arrivals) with reassignment (jobs can move)
Always use min{m,n} fastest servers (n = # jobs, m = # servers)
Clearing system without reassignment
Threshold policy is optimal: Use fastest available server j if n ≥ tj

6. Prior Work – The slow server problem
Clearing system with reassignment
Always use min{m,n} fastest servers
Clearing system without reassignment
Threshold policy: Use fastest j if n ≥ tj
Poisson arrivals with reassignment
Poisson arrivals without reassignment
Two servers: Threshold policy is optimal
More than 2 servers: Open

7. Our Results, with Usage Costs
Clearing system with reassignment
Threshold policy, but threshold and server preference order difficult to determine
Clearing system without reassignment
Threshold policy, server index = (h+βj)/μ j
Poisson arrivals, rate λ, with reassignment
Threshold policy, preference order depends on λ
Poisson arrivals without reassignment
2 servers: Thresholds, λ-dependent preference
More than 2 servers: Open

8. Our Results, with Usage Costs
Clearing system with reassignment
Threshold policy, but threshold and server preference order difficult to determine
Clearing system without reassignment
Threshold policy, server index = (h+βj)/μ j
Poisson arrivals, rate λ, with reassignment
Threshold policy, preference order depends on λ
Poisson arrivals without reassignment
2 servers: Thresholds, λ-dependent preference
More than 2 servers: Open

9. II. Clearing System, No Reassignment
n identical jobs with holding cost rate h
m servers with rate μj and usage cost βj
Jobs cannot be removed from processors
Theorem: Order servers in increasing order of
There exist thresholds, 1 = t1 ≤ t2 ≤ … ≤ tm, such that it is optimal to assign a job to server j if – j is the lowest indexed available server and – n ≥ tj

10. II. Clearing, No Reassignment – Computation
Theorem Details: Order servers so is increasing in j. Let There exist thresholds, 1 = t1 ≤ t2 ≤ … ≤ tm, such that it is optimal to assign a job to the lowest indexed server j if n ≥ tj where t1 = 1 and
regardless of the states of less preferred (higher indexed) servers.

11. Proof Outline (for all cases I-IV)
Define priorities for an individually optimal (IO) policy
Characterize the IO policy That is, show the IO policy has the given threshold structure
Show the IO policy is socially optimal (SO)

12. Proof Outline (for all cases I-IV)
Define priorities for an individually optimal (IO) policy
Assign arbitrary priorities to jobs in queue, available servers are offered to jobs in priority order
Job i is offered servers rejected by jobs 1,…,i-1
Job i decides to accept or reject the server to minimize its own total expected holding and usage cost
Characterize the IO policy
Show the IO policy is socially optimal (SO)

13. 2. Characterize IO Policy
Lemma: If there is only one server j, and the server is idle, the expected cost for job i (the i’th job in the queue) is
If the server is busy, it is
Corollary 1: (when i=1) Server j is preferred to server j+1 under the IO policy if

14. 2. Characterize IO Policy
Corollary 2: Under the IO policy with m servers, Server order: increasing order of
- Threshold policy: Job i will accept the lowest indexed available server if it is offered to it and if
where vi(j-1) is the expected cost for job i if it waits for one of the servers 1,…,j-1

15. 2. IO policy – Computation
Corollary 3:
That is, given job i is willing to accept any of servers 1,…,j and not servers >j and servers 1,…, j are all busy,
the expected cost for job i is the same as if it is the i+j’th job and there is a single super server with processing rate Mj and usage cost Bj.
This cost is independent of the policies of the other jobs (except that they will not use servers >j)

16. 2. IO policy – Computation
Corollary 3:
Corollary 3 and Corollary 2, which was that under IO, job i will accept the lowest indexed available server, j, if j is offered to i and if
give us…

17. 2. Thresholds for the IO policy
Corollary 4: The IO policy is a threshold policy: If j is the lowest indexed available server then job tj is assigned to server j if n ≥ tj, where t1 = 1 (the most preferred server is always used), and for j = 2,…,m,

18. 2. IO policy – “No recall”
Corollary 5: Under the IO policy, rejected servers will never be used
Under the IO policy, lower priority jobs have no effect on higher priority jobs
 Job n has no externality

19. Proof Outline (SO is a threshold policy)
Define priorities for an individually optimal (IO) policy ■
Show the IO policy is a threshold policy (we just did this) ■
Show the IO policy is SO (socially optimal)…

20. 3. IO = SO – Proof Idea Backwards Induction:
Deviating from IO for the first time step (with the lowest priority job, job n) and then following IO
hurts the lowest priority job, job n, and
has no effect on the other jobs
because job n has no externality
 Such a deviation cannot be SO

21. I. Clearing System With Reassignment
Can reassign jobs to new servers at any time
Now for the IO policy all jobs are labeled, and job i has preemptive priority over job i+1, and any job can leave a server at any time
That is, at any time all servers are offered to all jobs in order according to their priorities
IO is a threshold policy and IO = SO
Both easy; again job n imposes no externality

22. I. Clearing System With Reassignment
Can reassign jobs to new servers at any time
IO is a threshold policy and IO = SO
Determining server preference (and thresholds) is difficult
No longer (h+βj)/μ j
Server preference and thresholds are interdependent, and depend on all servers
Can generate preferences and thresholds recursively, using a “super server” result

23. I. Clearing, Reassignment: Computation
Let Π(j) be the j’th preferred server, i.e.,
the chosen server under the IO policy, if
servers Π(1),…,Π(j-1) all busy,
the other servers all idle, and
given some server is chosen
Let Πj = {Π(1),…,Π(j)}

24. I. Clearing, Reassignment: Computation
Let Π(j) be the j’th preferred server
Let Πj = {Π(1),…,Π(j)}
Recursively compute

25. I vs II (With/Without Reassignment)
If servers can be ordered so that and (faster = cheaper = smaller (h+β)/μ) then the thresholds can be related as follows: ti(no reassigment) = ti(reassignment) + ti(no reassigment, β=0) when ti(reassignment) > j

26. III. Arrivals and Reassignment
Theorem: With reassignment and Poisson arrivals, rate λ: IO = SO, a threshold policy
For IO policy: LCFP-P (Preemptive LCFP)
Higher priority (e.g., arriving) jobs can preempt jobs in service and can move to other servers
 lower priority jobs have no effect on higher priority jobs, i.e.,
Lowest priority job has no externality**
IO policy is a threshold policy – easy
IO = SO – same proof idea, using **

27. III. Arrivals and Reassignment
General structure (threshold policy) easy
Computation of the server preference order and thresholds is even more difficult
So suppose only 2 servers…

28. III. Arrivals, Reassignment, 2 Servers
If and
then server 1 is always preferred
If and then there exists λ0
such that server 1 is preferred iff λ < λ 0
The preferred server is always used
The threshold for the other server is easily computed

29. III. Arrivals and Reassignment
We use the IO LCFS-P policy to determine the thresholds, but
For the SO policy, when a job is assigned, it can be any job (need not be LCFS-P)
For example, FCFS with given tj’s is SO
Or we could assign jobs to minimize moves:
– Once a job is assigned to server j (when n ≥ tj) remove it from server j only if n drops below tj before the job finishes service

30. IV. Arrivals, No Reassignment, 2 Servers
Theorem: Without reassignment and two servers, IO = SO, a threshold policy:
The preferred server is always used
Server 2 is used when n ≥ t
Where the IO policy assumes LCFP-P
Higher priority jobs in queue can preempt lower priority jobs in service, but cannot leave servers
From a social perspective, jobs preempting other jobs have no effect (job swap) (indistinguishable jobs, exponential services)

31. IV. Arrivals, No Reassignment, 2 Servers
Theorem: Without reassignment and two servers, IO = SO, a threshold policy:
The preferred server is always used
Server 2 is used when n ≥ t
Where the IO policy assumes LCFP-P
IO policy is a threshold policy – easy
IO = SO
Same idea: Deviating from IO for the first decision with job n hurts job n and has no effect on others

32. IV. Arrivals, No Reassignment, 2 Servers
Theorem: Without reassignment and two servers, IO = SO, a threshold policy:
The preferred server is always used
Server 2 is used when n ≥ t
Again, the SO policy can be implemented with arbitrary job assignments, given the threshold
E.g., FCFS, without preemption

33. IV. Arrivals, No Reassignment, 2 Servers
If and then server 1 is
always preferred
If and then there exists
λ0 such that server 1 is preferred if λ < λ 0
In contrast to the reassignment case, we were unable to show that
If and then
server 2 is preferred if λ ≥ λ 0

34. IV. Arrivals, No Reassignment, >2 Servers
Theorem: Without reassignment and more than two servers, IO ≠ SO
Even with preemptive LCFP for the IO policy,
The decision of whether or not to assign the lowest priority job in queue to a server can impose an externality on jobs with even lower priorities on less preferred servers
The IO policy is still a threshold policy, but the threshold for a server depends on the relative priorities of jobs on less preferred servers

35. Summary
Theorem: The IO policy is a threshold policy, and IO = SO, except for the case:
Arrivals, no reassignment, and > 2 servers
Determination of the server preference order and thresholds are complicated, but they can be obtained by considering the IO policy
Showing that the IO policy is a threshold policy and that IO = SO gives an easy way to show the SO policy is a threshold policy