1. Queuing Theory For Dummies
Jean-Yves Le Boudec 1

2. All You Need to Know About Queuing Theory
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Queuing is essential to understand the behaviour of complex computer and communication systems
In depth analysis of queuing systems is hard
Fortunately, the most important results are easy
We will study only simple concepts 2

3. 1. Deterministic Queuing
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Easy but powerful
Applies to deterministic and transient analysis
Example: playback buffer sizing 3

4. Use of Cumulative Functions
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5. Solution of Playback Delay Pb
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A(t)
A’(t)
D(t)
time
bits
d(0)
d(0) - D
d(0) + D
(D1): r(t - d(0) + D)
(D2): r (t - d(0) - D)
d(t) A. 5

6. 2. Operational Laws Intuition:
Say every customer pays one Fr per minute present
Payoff per customer = R
Rate at which we receive money = N
In average λ customers per minute, N = λ R 6

7. Little Again
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Consider a simulation where you measure R and N. You use two counters responseTimeCtr and queueLengthCtr. At end of simulation, estimate R = responseTimeCtr / NbCust N = queueLengthCtr / T where NbCust = number of customers served and T=simulation duration
Both responseTimeCtr=0 and queueLengthCtr=0 initially
Q: When an arrival or departure event occurs, how are both counters updated ? A: queueLengthCtr += (tnew - told) . q(told) where q(told) is the number of customers in queue just before the event. responseTimeCtr += (tnew - told) . q(told) thus responseTimeCtr == queueLengthCtr and thus N = R x NbCust/T ; now NbCust/T is our estimator of  7

8. Other Operational Laws
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9. The Interactive User Model9 9

10. Network Laws 10

11. Bottleneck Analysis Apply the following two bounds Example (1) (2) 17
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Apply the following two bounds
Example
(1)
(2) 17 11

12. Throughput Bounds
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13. Bottlenecks
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14. DASSA
Intuition: within one busy period: to every departure we can associate one arrival with same number of customers left behind 14

15. 3. Single Server Queue 15

16. i.e. which are event averages (vs time averages ?)
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i.e. which are event averages (vs time averages ?) 16

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19. 16/04/2017 19

20. Non Linearity of Response Time20

21. Impact of Variability
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22. Optimal Sharing Compare the two in terms of Response time Capacity 22
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Optimal Sharing
Compare the two in terms of
Response time
Capacity 22

23. The Processor Sharing Queue
Models: processors, network links
Insensitivity: whatever the service requirements:
Egalitarian 23

24. PS versus FIFO PS
FIFO 24

25. 4. A Case Study
Impact of capacity increase ?
Optimal Capacity ? 25

26. Methodology 26

27. 4.1. Deterministic Analysis27

28. Deterministic Analysis28

29. 4.2 Single Queue Analysis
Assume no feedback loop: 29

30. 4.3 Operational Analysis A refined model, with circulating users
Apply Bottleneck Analysis ( = Operational Analysis )
Z/(N-1) -Z
1/c
waiting time 30

31. 31

32. 32

33. 5. Networks of Queues Stability
Queuing networks are frequently used models
The stability issue may, in general, be a hard one
Necessary condition for stability (Natural Condition) server utilization < at every queue 33

34. Instability Examples
Poisson arrivals ; jobs go through stations 1,2,1,2,1 then leave
A job arrives as type 1, then becomes 2, then 3 etc
Exponential, independent service times with mean mi
Priority scheduling
Station 1 : 5 > 3 >1
Station 2: 2 > 4
Q: What is the natural stability condition ?
A: λ (m1 + m3 + m5 ) < 1 λ (m2 + m4) < 1 34

35. If λ (m1 + … + m5 ) < 1 network is stable; why?
λ = 1 m1 = m3 = m4 = 0.1 m2 = m5 = 0.6
Utilization factors
Station 1: 0.8
Station 2: 0.7
Network is unstable !
If λ (m1 + … + m5 ) < 1 network is stable; why? 35

36. Bramson’s Example 1: A Simple FIFO Network
Poisson arrivals; jobs go through stations A, B,B…,B, A then leave
Exponential, independent service times
Steps 2 and last: mean is L
Other steps: mean is S
Q: What is the natural stability condition ?
A: λ ( L + S ) < 1 λ ( (J-1)S + L ) < 1
Bramson showed: may be unstable whereas natural stability condition holds

37. Bramson’s Example 2 A FIFO Network with Arbitrarily Small Utilization Factor
m queues
2 types of customers
λ = 0.5 each type
routing as shown, … = 7 visits
FIFO
Exponential service times, with mean as shown S S L S S L S S L S S L
Utilization factor at every station ≤ 4 λ S
Network is unstable for S ≤ 0.01 L ≤ S8 m = floor(-2 (log L )/L) 37

38. Take Home Message
The natural stability condition is necessary but may not be sufficient
There is a class of networks where this never happens. Product Form Queuing Networks 38

39. Product Form Networks Customers have a class attribute
Customers visit stations according to Markov Routing
External arrivals, if any, are Poisson
2 Stations
Class = step, J+3 classes
Can you reduce the number of classes ? 39

40. Chains
Customers can switch class, but remain in the same chain ν 40

41. Chains may be open or closed
Open chain = with Poisson arrivals. Customers must eventually leave
Closed chain: no arrival, no departure; number of customers is constant
Closed network has only closed chains
Open network has only open chains
Mixed network may have both 41

42. 3 Stations
4 classes
1 open chain
1 closed chain ν 42

43. Bramson’s Example 2 A FIFO Network with Arbitrarily Small Utilization Factor
2 Stations
Many classes
2 open chains
Network is open 43

44. Visit Rates 44

45. Visit rates θ11 = θ13 = θ15 = θ22 = θ24 = λ θsc = 0 otherwise
2 Stations
5 classes
1 chain
Network is open
Visit rates θ11 = θ13 = θ15 = θ22 = θ24 = λ θsc = 0 otherwise 45

46. ν 46

47. Constraints on Stations
Stations must belong to a restricted catalog of stations
See Section 8.4 for full description
We will give commonly used examples
Example 1: Global Processor Sharing
One server
Rate of server is shared equally among all customers present
Service requirements for customers of class c are drawn iid from a distribution which depends on the class (and the station)
Example 2: Delay
Infinite number of servers
No queuing, service time = service requirement = residence time 47

48. Example 3 : FIFO with B servers
FIFO queueing
Service requirements for customers of class c are drawn iid from an exponential distribution, independent of the class (but may depend on the station)
Example of Category 2 (MSCCC station): MSCCC with B servers
FIFO queueing with constraints
At most one customer of each class is allowed in service
Examples 1 and 2 are insensitive (service time can be anything) Examples 3 and 4 are not (service time must be exponential, same for all) 48

49. Say which network satisfies the hypotheses for product form
B (FIFO, Exp)
C (Prio, Exp) 49

50. The Product Form Theorem
If a network satisfies the « Product Form » conditions given earlier
The stationary distrib of numbers of customers can be written explicitly
It is a product of terms, where each term depends only on the station
Efficient algorithms exist to compute performance metrics for even very large networks
For PS and Delay stations, service time distribution does not matter other than through its mean (insensitivity)
The natural stability condition holds 50

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52. 52

53. Conclusions
Queuing is essential in communication and information systems
M/M/1, M/GI/1, M/G/1/PS and variants have closed forms
Bottleneck analysis and worst case analysis are usually very simple and often give good insights
Queuing networks may be very complex to analyze except if product form – be able to recognize them ! 53