Defining SteadyState Service Level Agreeability using SemiMarkov Process

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Defining Steady-State Service Level
Agreeability using Semi-Markov Process
Ranjith Vasireddy Kishor S. Trivedi rv5, kst
_at_ee.duke.edu Duke University

• June 27, 2006

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Motivation

• Service level agreements
• Ability of servers to deliver services to the
  clients according to the pre-defined agreements
• Assumption in this work service-level agreements
  are based on response times
• Should be valid (ultimately all predefined
  agreements should result in the improved
  response-times)
• Service level agreeability is not captured in
  the definition of availability

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Problem statement

• Guarantee of five-nines availability does not
  imply any guarantees to its service level
  agreements
• It is possible that system is available, but
  might not be satisfying the service level
  agreements
• Notion of service level agreeability need to be
• defined mathematically
• To compare service level agreements between two
• related systems

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Modeling using semi-Markov process

• Define the states by classifying the observed
  response-times
• into finite number of clusters
• Each cluster forms a state in semi-Markov process
• Four states indicate
• Highly SLA-satisfying (1)
• SLA-satisfying (2)
• SLA-violating (3)
• Highly SLA-violating (4)
• Reasons for more than two states
• To know whether failure is gradual or sudden
• If gradual depending on the system state at given
  time, we can predict the failure
• Otherwise, looking at system state at any given
  time may not be of
• much help

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Modeling using semi-Markov process

• Semi-Markov process allows general sojourn-time
  distributions
• CTMC would have allowed only exponential
  distribution
• Characterization of semi-Markov process
• Think of transitions as occurring in two stages
• First stage chain stays in state i for some
  amount of time described the sojourn time
  distribution Hi(t)
• Second stage Chain moves from state i to state j
  with a transition probability of pij

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Semi-Markov Process (SMP)

• Semi-Markov Process is characterized by
• Sojourn time distribution in each state
• Probability transition matrix (P) among the
  states
• Vector v indicates steady state probability of
  embedded DTMC
• we subject to normalization condition ?vi 1, to
  solve for vi
• ?i vihi/(?(vihi)) , hi is mean sojourn time
  in state i
• We get steady-state probability of state i in a
  SMP

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Example SMP (Experiments on MySQL server)

• Embedded DTMC
• For eg. P41 (41)/(4j) 2/6 1/3

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Experiments on MySQL server

• Steady-State Probabilities
• ?1 0.221732
• ?2 0.622514
• ?3 0.090908
• ?4 0.064317
• SLA satisfied if resp. time lt 3.5 seconds
• Steady-state SL Agree-ability ?1 ?2
• 0.8442
• Steady-state SL Un-Agreeability ?3 ?4
• 0.1552
• Theoretically, it implies that system does not
  satisfy SLA 1364.8 hrs / year
• this simple system can be easily shown to satisfy
  five-nines availability
• Varying threshold response times gives different
  SLA-bility values.

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References

• Kishor S. Trivedi. Probability and Statistics
  with Reliability, Queuing, and Computer Science
  Applications, second-edition, John Wiley, 2001.
• Kalyanraman Vaidyanathan, Kishor S. Trivedi A
  Comprehensive Model for Software Rejuvenation.
  IEEE Trans. Dependable Sec. Comput. 2(2)
  124-137 (2005)
• Ranjith Baghwan, S. Savage, G M Voelker.
  Understanding Availability. Proceedings of the
  2nd International workshop on Peer-to-Peer
  Systems (IPTPS03), Feb 2003.

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Problem statement

• System at any given point of time can be in one
  of the set of possible states (S1, S2, .. SN).
• In our definition, above set is divided into two
  mutually exclusive sets SLA satisfying states and
  SLA violating states
• Define SLA(T) of a system as the probability that
  the system is in one of the SLA satisfying
  states at time t.
• Often, we are interested steady-state values.
• Steady-state SLA as the limiting value of SLA(t)
  as t approaches infinity

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Experiments on MySQL server

• Experimentation on tables with 10 million records
  
• Each record has 3 int elements
• Each column can have values between 0,100)
• Each client executes insert, delete, range-query
  selected at random
• Each delete is
• Delete from table Y where a? and b?
• P(Aa Bb) (1/100)(1/100) 1/(104)
• Since table size is x106 records 10 102
  104
• Each such delete deletes 10100 records
  probabilistically
• Each insert inserts 10100 records to keep the
  table size roughly the same