Process Fundamentals, Queueing Theory,

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Process Fundamentals, Queueing Theory,
Simulation

• Stephen Blythe

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Process States

• A process state describes its current status
• running
• waiting
• for I/O to complete
• for an available CPU
• for system call to complete
• ... ???
• completed
• ... ???

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Process State Transitions
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Process State Transitions ...
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Process State Transitions ...
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Process State Transitions ...
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Process States O/S Design

• Any change in am O/S state means an O/S change
• code modification
• performance changes
• extra hardware needed now?
• ...
• Can either just build the new O/S (costly) ...
• ... or simulate the new system (cheaper), giving
• performance measures
• prediction of hardware needs
• ...
• ... or ...

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Queuing Theory

• ... OR , perform queuing analysis
• formal, mathematical analysis of states and
  queues
• under assumptions, gives accurate results
• no need for any coding whatsoever!
• but, one needs to understand the math involved

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Queuing Theory ...

• Consider the following simple queuing system
• given that
• the system is stable (the queue is finite)
• ?in is job arrival rate (new jobs per sec.)
• ?out is job completion rate (jobs finished per
  sec.)
• Can we relate ?in to ?out ?

CPU
?in ?out (Surprise!!)
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Queuing Theory ...

• Other quantities of interest
• Ts average service time (time in CPU)
• Tw average wait time (time spent in CPU queue)
• w average number waiting jobs (CPU queue
  length)
• ? average server utilization (CPU utilization)
• r average number of jobs in system
• Tr average response time
• Some simple observations/formulas
• TrTsTw
• ??inTs
• r?inTr (Littles formula/law)

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Queuing Theory ...

• since TrTs/(1-?) and r?inTr ,
• r?inTs/(1-?)
• (?inTs)/(1-?)
• ?/(1-?) ?
• Can you show that w?2/(1-?) ?
• How about Tw?Ts/(1-?) ?

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O/S Example
?ts
?in
?out
Ts

• What does adding a time slice do?
• Suppose
• ?in 1 job / 60 sec
• Ts 20 sec/job
• Prob(?ts) Prob(?out)

0.5
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O/S Example ...
?ts
?in
?out
Ts
?in ?ts

• ?out ?in
• ?out 0.5 (?in ?ts)
• ?ts 0.5 (?in ?ts)
• 0.5 ?ts 0.5 ?in
• ?ts ?in

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O/S Example ...
?ts
?in
?out
Ts
?in ?ts

• But, theres a catch
• derived equations assume only 1 pass through
  CPU
• this model has several
• how long is typical pass through CPU?
• Assume it is 10 seconds here (well discuss
  later)
• now, have 2 passes through CPU with Ts10
• and Tr 2 (10/(1-?)) 210/(1-1/3)30 (secs.)

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O/S Example ...
?ts
?in
?out
Ts
?in ?ts

• Another catch dont know probabilities in
  advance
• Do know time time slice length, though!!
• ex. Ts30s , time slice5s, ?in1job/100seconds
• over n secs, exitsn/30, time slicesn/5 (at
  most)
• P(exit) exits/(exits time slices)
• P(time slice) time slices(exits time
  slices)
• time in CPU P(event)(event length)

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O/S Example ...
?ts
?in
?out
Ts
?in ?ts

• P(exit)limn?8(n/30)/( (n/30)(n/5) ) 1/7
• P(time slice)limn?8(n/5)/( (n/30)(n/5) ) 6/7
• again Ts30s , time slice5s, ?in1job/100seconds
  
• ?ts6/7(?in?ts), so ?ts6?in
• ?(?in?ts)Ts7?inTs7(1/100)(30/7)3/10
• Tr6 passes (Ts/(1-?)) 65/(1-3/10)300/7

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Another Option

• As systems get larger and more complex ...
• ... Queuing Analysis becomes more difficult
• ... sometimes too difficult.
• Instead, perform a discrete event simulation
• simulates large amounts of system time (years)
• ... taking only seconds of real time
• ... calculates important statistics along the
  way
• Tr, ? (for all devices in system), ...

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Discrete Event Simulation

• The core of Discrete Event Simulation is the
  event
• In O/S, an event is an action that needs O/S
  intervention
• new job arriving
• time slice expiring
• job requires I/O
• job completes
• I/O completes
• ...

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Discrete Event Simulation ...

• Basic Idea
• start with a single seed event (process
  arrival)
• add the event to a sorted event queue
• set curr_time0
• while(queue not empty curr_timeltmax_time)
• remove first event ev from queue
• set curr_timeev.time
• update internal statistics
• generate next event(s) based on this event
• add the new events to the queue

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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• seed event 1st process enters

Statistics curr_time 0 jobs_done 0
CPU_time 0
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process event (0,0, New)
• adds (0,75, Exit) and ...

Statistics curr_time 0 jobs_done 0
CPU_time 0
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process event (0,0, New)
• adds (0,75, Exit) and ...
• (1, 100, New)

Statistics curr_time 0 jobs_done 0
CPU_time 0
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process event (0,75, Exit)
• updates stats

Statistics curr_time 75 jobs_done 1
CPU_time 75
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process event (1,100, New)
• adds (1, 175, Exit) and
• (2,200, New)

Statistics curr_time 100 jobs_done 1
CPU_time 75
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process event (1,175, Exit)
• updates stats

Statistics curr_time 175 jobs_done 2
CPU_time 150
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Discrete Event Simulation ...

• When done, will have
• total time in CPU
• total time so far
• number of jobs completed
• could also track other stats along the way
• but there are problems
• real jobs dont have constant CPU (or arrival)
  times
• vary values using exponential random distribution
• CPU queue not used in prior example

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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• seed event 1st process enters

CPU Queue
CPU Status
idle
Statistics curr_time 0 jobs_done 0
CPU_time 0
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process event (0, 0, New)
• generate CPU time 89 ns
• generate next new 08686 ns

CPU Queue
0
CPU Status
idle
Statistics curr_time 0 jobs_done 0
CPU_time 0
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• check CPU - is it busy?
• yes, so pull a job from queue
• schedule CPU exit event

CPU Queue
CPU Status
busy
Statistics curr_time 0 jobs_done 0
CPU_time 0
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process next event (1, 86, New)
• CPU busy, so add to queue
• generate next new 86112198

CPU Queue
1
CPU Status
busy
Statistics curr_time 86 jobs_done 0
CPU_time 86
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process next event (0, 89, Exit)
• Update Statistics
• Schedule queued job (t8951)

CPU Queue
CPU Status
busy
Statistics curr_time 89 jobs_done 1
CPU_time 89
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process next event (1, 140, Exit)
• Update Statistics
• No jobs in queue ...

CPU Queue
CPU Status
idle
Statistics curr_time 140 jobs_done 2
CPU_time 140
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process next event (2, 198, New)
• queue empty, run on CPU (exit event)
• generate next new event

CPU Queue
CPU Status
busy
Statistics curr_time 198 jobs_done 2
CPU_time 140
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Discrete Event Simulation ...

• Example - two state model
• jobs arrive every 100 ns
• takes 75 ns to serve them
• process next event (2, 276, Exit)
• update statistics
• queue empty ...

CPU Queue
CPU Status
busy
Statistics curr_time 276 jobs_done 3
CPU_time 218