CSC 8420 Advanced Operating Systems Georgia State University Yi Pan

1

Distributed Process Scheduling
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
2

Issues in Distributed Scheduling

1. The effect of communication overhead
2. The effect of underlying architecture
3. Dynamic behavior of the system
4. Resource utilization
5. Turnaround time
6. Location and performance transparency
7. Task and data migration

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
3

Classification of Distributed Scheduling
Algorithms

1. Static or off-line
2. Dynamic or on-line
3. Real-time

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
4

Process or Load Modelling

1. Precedence graph
2. Communication graph
3. Disjoint process model
4. Statistical load modeling

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
5

System Modelling

1. Communication system model
2. Processor pool model
3. Isolated workstation model
4. Migration workstation model

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
6

Optimizing Criteria

1. Speedup
2. Resource utilization
3. Makespan or completion time
4. Load sharing and load balancing

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
7

Static Scheduling
Statistical load, isolated workstation
l
m
Turnaround time1/(m-l)
l
m
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
8

Static Scheduling
Statistical load, processor pool
m
2l
m
Turnaround timem/((m-l)(ml))
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
9

Static Scheduling
Communication process model, communication system
model
The general problem is NP-complete.

1. Stones algorithm (2 heterogeneous processors,
   arbitrary communication process graph).
2. Bokharis algorithm (n homogeneous processors,
   linear communication process graph, linear
   communication system)

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
10

Stone's Algorithm
Assumptions

1. Two heterogeneous processors. Therefore, the
   execution cost of each process depends on the
   processor. Further, the execution cost for each
   process is known for both processors.
2. Communication cost between each pair of processes
   is known.
3. Interprocess communication incurs negligible cost
   if both the processes are in the same processor.

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
11

Stone's Algorithm
Objective Function
Stones Algorithm minimizes the total execution
and communication cost by properly allocating the
processors among the processes.
Let, G(V,E) be the communication process model.
Let A and B be two processors. Let wA(u) and
wB(u) be the cost of executing process u on
processors A and B respectively. Let, c(u,v) be
the cost of interprocess communication between
processes u and v if they are allocated different
processors. Further, S be the set of processes to
be executed on processor A and V-S be the set of
process to be executed on processor B. Stones
algorithm computes S such that the objective
function Su e S wA(u)Sv e V-S wB(v)Su e S, v e
V-S c(u,v) is minimized.
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
12

Stone's Algorithm
Example
1 2 3 4
1 4 5 0
2 1 2
3 2
4
Processes Cost at Processor A Cost at Processor B
1 3 2
2 1 5
3 2 4
4 3 3
Computation Costs
Communication Costs
If process 1 and 2 are allocated to processor A,
and the rest in processor B, then the total cost
is 3143501219.
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
13

Stone's Algorithm
Commodity-flow problem
Stones algorithm reduces the scheduling problem
to the commodity-flow problem described below.
Let G(V,E) be a graph with two special nodes u
(source) and v (sink). Each edge has a maximum
capacity to carry some commodity. What is the
maximum amount of the commodity that can be
carried from the source to the sink.

1. There is a known polynomial time algorithm to
   solve the commodity-flow problem
2. Let S be a subset of V such that the source is
   in S and the sink is in V-S. A set of edges (say
   C) with one end in S and the other end in V-S is
   called a cut and the sum of the capacities of the
   edges in C is called the weight of the cut. It
   can be shown that the maximum-flow is equal to
   the minimum possible cut-weight. This is called
   max-flow, min-cut theorem.

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
14

Stone's Algorithm
Reduction to the commodity-flow problem
Stones algorithm reduces the scheduling problem
to the commodity-flow problem as follows.
Let G(V,E) be the communication process graph.
Construct a new graph G(V,E) by adding two
new nodes a (source, corresponding to processor
A) and b (sink, corresponding to processor B).
For every node u in G, add the edges (a,u) and
(b,u) in G. The weights (capacities) of the new
edges will be wB(u) and wA(u) respectively.

1. Note that a cut in G gives a processor
   allocation for the job.
2. Further, the weight of the cut is same as the
   cost (objective function) of execution cost of
   communication for the processor allocation given
   by the cut.
3. Therefore, if we compute the max-flow on G, the
   corresponding min-cut gives the best processor
   allocation.

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
15

Stone's Algorithm
5
Example
3
2
1
4
1
2
2
1
B
A
5
4
3
2
2
4
3
3
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
16

Bokhari's Algorithm
Assumptions

1. n homogeneous processors, connected in a linear
   topology.
2. kgtn processes, linear communication graph.
3. Communication links are homogeneous.
4. Computation and communication costs are known.
5. Two processes not communicating directly will not
   be allocated the same processor.
6. Two communicating processes, if not allocated
   same processor, must be in the adajacent
   processors.

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
17

Bokhari's Algorithm
Objective function
Bokharis algorithm minimizes the sum of the
total communication cost and the maximum
execution cost in a processor. Note the
difference with the objective function in Stones
algorithm.
Let P1,,Pn be processors in linear order and
p1,,pk be processes in linear order. Let the
execution cost of pi be wi and the communication
cost between pi and pi1 be ci. If processes
pj(j) to pi(j1)-1 be allocated to processor Pi,
then the objective function for Bokharis
algorithm will be maxiWjSi1ncj(i) where Wj is
the execution cost for processor j.
Wjwi(j)wj(i1)-1.
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
18

Bokhari's Algorithm
Example
2
4
2
1
3
3
2
4
1
2
1
P1
P2
P3
The value of the objective function
max(32,4,121)(21)8.
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
19

Bokhari's Algorithm
To solve the problem Bokhari constructed the a
layered graph. The graph has n2 layers, where n
is the number of processors. The top and bottom
layer has 1 node each. All other layers have k
nodes where k is the number of processes. We
number the layers from 0 thru k1 and denote the
ith node of jth layer by v(i,j). The node in the
top (0th) layer is adjacent to all nodes in layer
1. The node in the bottom layer is adjacent to
all nodes in the kth layer. Other v(i,j)s are
adjacent to v(i,j1),,v(k,j1).
Example of a layered graph constructed for n2
and k3. Note that any path from the top layer to
the bottom layer gives a processor allocation
subject to the restrictions described earlier and
vice versa.
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
20

Bokhari's Algorithm
Each edge e in the graph has two weights, w1(e)
and w2(e). If e is an edge connecting v(i,j) and
v(r,j1) then w1(e) is the total execution time
of the processes pi to pr-1 and w2(e) is the
communication cost between pr-1 and pr. Note
that, a shortest path from the top to bottom on
the basis of w2 will minimize the total
communication cost. Any known shortest path
algorithm may be used to do that. Further, w1 may
take only certain values. To be precise, there
are k1C2 possible values of w1. We can compute
those values and sort them. Then we can restrict
the w1 by deleting all edges with w1(e) greater
than the restricted value. A shortest path in the
new layered graph will minimize the total
communication cost subject to the restriction
that the maximum of the execution cost in any
processor will not exceed the threshold value.
Using this method we can compute the objective
function for each of the possible k1C2 threshold
values. The best processor allocation can thus be
computed. Note that a clever binary search over
the possible values of w1 may reduce the
computational complexity of the algorithm
significantly.
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
21

Dynamic Scheduling
The Goal of dynamic scheduling is to load share
and balance dynamically. This goal is achieved by
migrating tasks from heavily loaded processors to
lightly loaded processors.

• Two types of task migration algorithms
• Sender-initiated
• - Transfer policy (when to transfer?)
• - Selection policy (which task to transfer?)
• - Location policy (where to transfer?)
• Potentially unstable.
• Receiver-initiated

CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
22

Real-time Scheduling
Real-time Scheduling
Definitions
A real-time task consists of Si, the earliest
start time, Di, the deadline to finish the task
and Ci, the worst-case completion time.
A set of real-time tasks is called a real-time
task set or a real-time system.
A system of real-time tasks is called hard
real-time system if all of the tasks in the
system must complete execution within their
respective deadlines. Even if one task misses its
deadline, the whole system is considered
incorrectly executed.
A system of real-time tasks is called soft
real-time system if the performance is judged by
how many tasks missed deadline and by how much.
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
23

Real-time Scheduling
Real-time Scheduling
Definitions
A schedule for a real-time system is a CPU
assignment to the tasks of the system such that a
CPU is assigned to no more than one task at one
time.
A schedule for a real-time system is called valid
if no task is assigned CPU before its
earliest-start time and all tasks complete
execution within their respective deadlines.
A real-time system is called feasible if there is
a valid schedule for the system.
CSC 8420 Advanced Operating Systems
Georgia State University
Yi Pan
24
Real-Time Scheduling

• Real-time task TASKi(Si, Ci, Di)
• Si earliest possible start time
• Ci worst-case execution time
• Di deadline
• Set of tasks V TASKi I 1, ., n

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• Aperiodic arrive at arbitrary time
• Periodic arrival time, execution time and
  deadline are predictable
• V Ji (Ci, Ti)
• Where Ci is the execution time and Ti is the
  period time.

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• A schedule is a set of execution intervals
• A (si, fi, TASKi) i 1, . n
• si actual start time
• fi actual finish time
• TASKi the task executed in the time interval

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• Valid Schedule if
• For every i, si lt fi
• For every i, fi lt si1
• If TASKi k, then Sk lt si and fi lt Dk (within
  its time interval)

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Rate Monotonic Scheduling

• Assume t is the request time for task TASKi
• TASKi can meet its deadline if the time spent
  executing higher-priority tasks during the time
  interval (t,tDi) is Di-Ci or less.
• Critical instant for task TASKi occurs when task
  TASKi and all higher-priority tasks are scheduled
  simultaneously.

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• If task TASKi can meet its deadline at a critical
  instant, then it cal always meet its deadline.
• Considering schedulability of task TASKl if we
  move the request time of higher-priority tasks
  TASKh slightly forward.
• Might reduce the time spent executing TASKh in
  the interval (t,tDTASKl), but not increasing .

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• Considering schedulability of task TASKl if we
  move the request time of higher-priority tasks
  TASKh slightly backward.
• Might reduce the time spent executing TASKh in
  the interval (t,tDTASKl), but not increasing .
• Hence, at the critical instant, the amount of
  time spent by high-PR tasks is maximized.

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• We can now determine if a priority assignment
  results in a feasible schedule by simulating the
  execution of the tasks at the critical instant of
  the lowest priority task.
• If the lowest-PR task can meet its deadline
  starting from its critical deadline, all tasks
  will.

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• Tasks with short periods have less time in which
  to complete their execution than tasks with
  longer deadlines.
• Hence, give them higher priority.
• Rate Monotonic Priority Assignment
• If Th lt Tl then PRh gt PRl

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• If there is a static PR assignment such that any
  resulting schedule is feasible, the Rate
  Monotonic PR Assignment policy will produce
  feasible schedule.
• It is optimal in this sense.
• How to prove? Using exchange policy for TASKi and
  TASKi1.