In this study, we obtain an exact representation of the distribution of the maximum order statistic for homogeneous and heterogeneous random variables that have a matrix–exponential (ME) representation, from which the mean and higher moments follow. We can then apply these results to M/G/1, M/G/1/N, M/G/1//N, M/G/C, and G/G/1//N queues where the stationary queue length and response time distributions, and other performance measures can be ascertained for a wide class of split-merge queues. We give examples of subtasks with homogeneous and heterogeneous service time distributions, subtask failure/repair, G/C-type systems, and a variable number of forked subtasks.

2. Exact Analysis of Some
Split-Merge Queues
Lester Lipsky
Computer Science & Engineering
University of Connecticut
Pierre Fiorini
CF Search
Portsmouth, NH

3. Contents
• Introduction
– Synchronized Queues
• Fork-Join Queues
• Split-Merge Queues
• Related Work
– Research Strategies
– Harrison & Zertal (2003)
– Lebrecht & Knottenbelt (2007)
– Tsimashenka & Knottenbelt (2014)
• Background
– Order Statistics
• Homogenous
• Heterogeneous
– Matrix Exponential Distributions
• Matrix Exponential Functions
• Homogenous Order Statistics
• Heterogeneous Order Statistics
• Joint Distributions
– M/G/1 Queues
• PK Formula
• Stationary Queue Length Distribution
• Response Time Distribution
• Examples of Split-Merge Queues
– Homogeneous Split–Merge Queues
– Heterogeneous Split–Merge
Queues
– Split–Merge Queues with Subtask
Failure & Repair
– Split–Merge Queues with a
Variable Number of Subtasks

4. Introduction – Split-Merge Queues
• On arrival a job is split into n sub-tasks which are serviced
in parallel.
• Only when all the tasks finish servicing and have rejoined
can the next job start
• A type of “synchronized” queue

5. Introduction – Fork-Join Queues
• Incoming jobs are split on arrival for service by numerous servers
and joined before departure
• Jobs can arrive at any time
• Exact results only exist for N = 2
𝑇 =
12 − 𝜌
8
𝑥
1 − 𝜌
Nelson, R. & Tantawi, A. N. (1988).

6. Introduction – Difference Between
Fork-Join & Split-Merge Queues
• FJ – new jobs can arrive at any time
• SM – new jobs can only arrive after the last subtask finishes service
• This means SM provide upper-bound response time for FJ queues
• SM queues are “more synchronized”

7. • Most researchers have used the EMOS (Expected
Maximum Order Statistic) technique for the service time
distribution
• Usually computed via numerical integration
• They have applied this to the M/G/1 queue and used the
PK Formula to generate performance measures
Related Work – Research Strategies
𝑅 ~
E[max 𝑋1, 𝑋2, … , 𝑋 𝑛
2]
1 − 𝜌

8. Related Work - Harrison & Zertal (2003)
• “Queueing Models with
Maxima of Service Times”
• Found recursive way to
compute the moments of
o.s.
– Identical + Non-Identical
– Exact for exponential
– Approximate for non-
exponential
• Applied results to analyze
performance of RAID
subsystems (exponential)
for M/G/1 queues

9. Related Work - Lebrecht & Knottenbelt (2007)
• “Response Time
Approximations in Fork-
Join Queues”
• A response time
approximation of the fork-
join queue is presented
using EMOS
• Used to model performance
of RAID
• Used various distributions
for M/G/1 queues &
heterogeneous servers

10. Related Work – Tsimashenka & Knottenbelt (2014)
• “Trading off Subtask Dispersion
and Response Time in Split-
Merge Systems”
• Describe a methodology for
managing the trade off between
subtask dispersion and task
response time
• That is, the time between the
arrival of the first and last
subtasks originating from a given
task in the output buffer
– The “Range”
• Used M/G/1 queue to generate
performance measures

11. Contributions
Research Queueing
Models
Homogenous
Order Statistics
Heterogeneous
Order Statistics
Stationary
Queue Length
Distribution
Response Time
Distribution
Unreliable
SM Queues
Variable
Subtasks
Harrison &
Zertal (2003)
M/G/1
(PK
Formula)
Exact for
Exponential,
Approximations
Exact for
Exponential,
Approximations
No results No results No results No results
Lebrecht &
Knottenbelt
(2007)
M/G/1
(PK
Formula)
Numerical
Integration
Approximations &
Numerical
Integration
No results No results No results No results
Tsimashenka &
Knottenbelt
(2014)
M/G/1
(PK
Formula)
Not discussed Numerical
Integration
No results No results No results No results
Fiorini &
Lipsky (2015)
M/G/1,
M/G/1/N,
M/G/1//N,
M/G/C,
M/G/C//N,
G/G/1//N
Markov Chain Markov Chain Explicit ME
representation
Explicit ME
representation
ME rep ME rep

12. Background – Homogenous Order Statistics
F(x)
X1
X2
X3
X4
X5
X(1)≤ X(2) ≤ X(3) ≤ X(4) ≤ X(5)
Randomly sample F(x)
Order sample by size…
Parent Distribution
o.s. Distributions

13. Background – Homogenous Order Statistics

14. Background – Homogenous Order Statistics
• Interested in the Extremes
• Maximum
• Minimum

15. Background – Homogenous Order Statistics

16. Background – Heterogeneous Order Statistics
F1(x)
X1
X2
X3
X4
X5
X(1)≤ X(2) ≤ X(3) ≤ X(4) ≤ X(5)
Randomly sample Fi(x)
Order sample by size…
F2(x)
Fn(x)
Parent Distribution
Non-Identical o.s.
Distributions

17. Background – Heterogeneous Order Statistics
Let X1, X2, X3, … be the order statistics
The joint distribution is (Bapet & Beg 1989)

18. Background – LAQT
Any arbitrary pdf can be represented by an m-dimensional
vector-matrix pair < p, B >
Let X be a matrix-exponential (ME) r.v. greater than or
equal to 0. The cdf is
Let density function is
Moments can be computed by

19. Background – LAQT (M/G/1)
For the open M/G/1 queue, the stationary queue length
probabilities are given by
where

20. Background – LAQT (M/G/1)
For the open M/G/1 queue, the mean queue length can be
calculated by
or

21. Background – LAQT (M/G/1)
For the open M/G/1 queue, the response time distribution
can be calculated by

22. • Intuition…
– Think of 4 tasks (i.e., “r.v.’s”) running concurrently…
• The 1st task finishes at t1
• The 2nd finishes at t2
– The maximum happens at X(4)
ME Order Statistics - Max
X(1)
X(2)
X(3)
X(4)
t1 t2 t3 t4

23. ME Order Statistics
State Transition Diagram
max{B1, B2, B3, B4} 𝑩1 ⊕ 𝑩2 ⊕ 𝑩3 ⊕ 𝑩4
𝑩1 ⊕ 𝑩2 ⊕ 𝑩3 𝑩1 ⊕ 𝑩2 ⊕ 𝑩4 𝑩2 ⊕ 𝑩3 ⊕ 𝑩4
𝑩1 ⊕ 𝑩2 𝑩3 ⊕ 𝑩4𝑩1 ⊕ 𝑩3 𝑩1 ⊕ 𝑩4 𝑩2 ⊕ 𝑩3 𝑩2 ⊕ 𝑩4
𝑩1 𝑩2 𝑩3 𝑩4
𝑩4
𝑩3 𝑩3
𝑩3
𝑩2 𝑩3
𝑩4 𝑩2
𝑩1
𝑩2
𝑩3 𝑩4
𝑩2
𝑩2 𝑩2
𝑩1
𝑩1
𝑩1𝑩3
𝑩3
𝑩3
𝑩4 𝑩4 𝑩4
𝑩1 ⊕ 𝑩3 ⊕ 𝑩4
𝑩1
𝑩4
𝑩3
𝑩1

24. ME Order Statistics – Markov Chain
max{B1, B2, B3, B4}
𝑩 =

25. ME Order Statistics - Max
Let X1, X2,…, Xn be independent (and possibly non-identical)
ME distributed r.v.’s having CDF

26. ME Order Statistics - Max

27. ME Order Statistics - Max
Need to construct the M & P matrices…

28. ME Order Statistics - Max

29. ME Order Statistics - Max

30. • Examples…
– Homogeneous split-merge queues
– Heterogeneous split-merge queues
– Split-Merge queues with unreliable subtasks
– Variable number of subtasks
Examples of Split-Merge Queues

31. • In this example, all subtasks are iid and have
the same parent distribution, 𝐹
• For now, we assume the following:
– n = 2 subtasks (but, n can by any number)
– Subtasks are exponentially distributed with
parameter m
Homogeneous Split-Merge Queues

32. Homogeneous Split-Merge Queues
• Process rate matrix
• Service time matrix
• Density function, which
is maximum of 2 exp r.v.
with rate µ

33. Homogeneous Split-Merge Queues
• Mean response time (using
the PK Formula for M/G/1
queues)
• Response time distribution (pdf)

34. Split-Merge vs. Fork-Join (n = 2)
(Nelson & Tantawi, 1988)
(Fiorini & Lipsky, 2015)
Mean upper-bound for FJ Queue, n = 2

35. • Upper-Bound Response Time Distribution for
Fork-Join queues where n = 2
Split-Merge vs. Fork-Join (n = 2)

36. • Here we assume all subtasks are iid and have
different parent distributions, 𝐹𝑖
• We assume the following:
– n = 2 subtasks (but, n can by any number)
– Subtasks are exponentially distributed, where
𝜇1 ≠ 𝜇2
Heterogeneous Split-Merge Queues

37. Heterogeneous Split-Merge Queues
• Process rate matrix
• Service time matrix
• Density function of
maximum o.s. of 2 non-
identical exponential
r.v.’s

38. Heterogeneous Split-Merge Queues
• Mean response time (using the PK Formula) for
M/G/1 queue

39. • Suppose a subtask fails at rate a and is
repaired at rate b
• Assume when subtask is in upstate, it
completes at rate m
• The generator of this process is
Unreliable Split-Merge Queues

40. • For n = 2, we have
Unreliable Split-Merge Queues

41. • The mean response time turns out to be (using the PK
formula) for the M/G/1 queue
Unreliable Split-Merge Queues

42. • Other modeling factors…
– Subtasks can have different failure rates, 𝛼𝑖
– Subtasks can have different repair rates, 𝛽𝑖
– Subtasks can be a mixture of subtasks that fail and
those that don’t
– Subtasks can have different recovery polices
• prd, prs, etc.
Unreliable Split-Merge Queues

43. Split-Merge Queues with Variable
Number of Forked Subtasks

44. • There is an a1p1 vector probability of 1 task being forked, a2p2
vector probability of 2 tasks, and so on…
Split-Merge Queues with Variable
Number of Forked Subtasks

45. Summary
Research Queueing
Models
Homogenous
Order Statistics
Heterogeneous
Order Statistics
Stationary
Queue Length
Distribution
Response Time
Distribution
Unreliable
SM Queues
Variable
Subtasks
Harrison &
Zertal (2003)
M/G/1
(PK
Formula)
Exact for
Exponential,
Approximations
Exact for
Exponential,
Approximations
No results No results No results No results
Lebrecht &
Knottenbelt
(2007)
M/G/1
(PK
Formula)
Numerical
Integration
Approximations &
Numerical
Integration
No results No results No results No results
Tsimashenka &
Knottenbelt
(2014)
M/G/1
(PK
Formula)
Not discussed Numerical
Integration
No results No results No results No results
Fiorini &
Lipsky (2015)
M/G/1,
M/G/1/N,
M/G/1//N,
M/G/C,
M/G/C//N,
G/G/1//N
Markov Chain Markov Chain Explicit ME
representation
Explicit ME
representation
ME rep ME rep

46. • Study performance bounds of homogenous and
heterogeneous fork-join queues
– Use split-merge queues
– Heavy-Tails?
– Not well understood
• Unreliable split-merge queues
– Subtasks can have different failure rates, 𝛼𝑖
– Subtasks can have different repair rates, 𝛽𝑖
– Subtasks can be a mixture of subtasks that fail and those that
don’t
– Subtasks can have different recovery polices
• prd, prs, etc.
Future Work

47. Questions???