Finite-Source Queueing Systems and their Applications

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János Sztrik 2001/08/05 Numerical Methods • A recursive method for the M/G/1/ system ⋆ The mathematical model ⋆ Numerical results Finite-Source Queueing Systems and their Applications
János Sztrik 2001/08/05 A recursive method for the M/G/1 system Closed-form solutions for the steady-state probabilities are very seldom. Different analytical methods are used to investigate the involved processes and related numerical problems. For the most common procedures and tools the interested reader is referred to [8, 24, 31, 33, 34, 49, 54, 60, 77, 81] In the following the results of [31] are introduced and some numerical examples are demonstrated. For ready use of numerical results a book of tables has been produced by [53] for M/M/r model. A collection of various theoretical results along with numerical work can also be found in [12] where he has discussed in detail its application in construction and mining. Takács [74] gives the explicit expression for the distribution of the number of working (up) machines of the (M/G/1) model. However, for a large number of machines the computation of probabilities using Theorem 2 in [74] (p. 195) may pose problem as it involves many factorials. Even for the simple model M/M/C, Gross and Harris [29] (p. 108) makes similar comments and proposed a recursive method for computing probabilities. To obtain the Finite-Source Queueing Systems and their Applications
Page 1 and 2: Finite-Source Queueing Systems and
Page 3 and 4: János Sztrik 2001/08/05 Introducti
Page 5 and 6: János Sztrik 2001/08/05 distributi
Page 7 and 8: János Sztrik 2001/08/05 Example 1
Page 9 and 10: János Sztrik 2001/08/05 Example 3
Page 11 and 12: János Sztrik 2001/08/05 Kendall’
Page 13 and 14: János Sztrik 2001/08/05 Kendall no
Page 15 and 16: János Sztrik 2001/08/05 job that w
Page 17 and 18: János Sztrik 2001/08/05 probabilit
Page 19 and 20: János Sztrik 2001/08/05 in accorda
Page 21 and 22: János Sztrik 2001/08/05 Performanc
Page 23 and 24: János Sztrik 2001/08/05 Hence we h
Page 25 and 26: János Sztrik 2001/08/05 will inclu
Page 27 and 28: János Sztrik 2001/08/05 Asymptotic
Page 29 and 30: János Sztrik 2001/08/05 Heterogene
Page 31 and 32: János Sztrik 2001/08/05 throughput
Page 33 and 34: János Sztrik 2001/08/05 If P (i) d
Page 35 and 36: János Sztrik 2001/08/05 The main a
Page 37 and 38: János Sztrik 2001/08/05 distributi
Page 39 and 40: János Sztrik 2001/08/05 rates λn
Page 41 and 42: János Sztrik 2001/08/05 (N − n)
Page 43 and 44: János Sztrik 2001/08/05 E[L] = N +
Page 45 and 46: János Sztrik 2001/08/05 7. Mean wa
Page 47: János Sztrik 2001/08/05 ρ N r Us
Page 51 and 52: János Sztrik 2001/08/05 The mathem
Page 53 and 54: János Sztrik 2001/08/05 � ∂
Page 55 and 56: János Sztrik 2001/08/05 Using (45)
Page 57 and 58: János Sztrik 2001/08/05 Setting s
Page 59 and 60: János Sztrik 2001/08/05 To get P
Page 61 and 62: János Sztrik 2001/08/05 Now from (
Page 63 and 64: János Sztrik 2001/08/05 Hence from
Page 65 and 66: János Sztrik 2001/08/05 or P ∗ 1
Page 67 and 68: János Sztrik 2001/08/05 Numerical
Page 69 and 70: János Sztrik 2001/08/05 H 2 (µ 1
Page 71 and 72: János Sztrik 2001/08/05 Finite-Sou
Page 73 and 74: János Sztrik 2001/08/05 N M E 10 D
Page 75 and 76: János Sztrik 2001/08/05 Preliminar
Page 77 and 78: János Sztrik 2001/08/05 〈αm〉,
Page 79 and 80: János Sztrik 2001/08/05 Theorem 1.
Page 81 and 82: János Sztrik 2001/08/05 The queuei
Page 83 and 84: János Sztrik 2001/08/05 i.e., the
Page 85 and 86: János Sztrik 2001/08/05 Proof. Let
Page 87 and 88: János Sztrik 2001/08/05 pε[(i1, i
Page 89 and 90: János Sztrik 2001/08/05 This agree
Page 91 and 92: János Sztrik 2001/08/05 (63),(64),
Page 93 and 94: János Sztrik 2001/08/05 substituti
Page 95 and 96: János Sztrik 2001/08/05 Consequent
Page 97 and 98: János Sztrik 2001/08/05 Performanc
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János Sztrik 2001/08/05 holding ti
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János Sztrik 2001/08/05 Mean delay
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János Sztrik 2001/08/05 Average nu
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János Sztrik 2001/08/05 Numerical
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János Sztrik 2001/08/05 N = 5 N =
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János Sztrik 2001/08/05 References
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János Sztrik 2001/08/05 [11] Bunda
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János Sztrik 2001/08/05 [23] Gelen
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János Sztrik 2001/08/05 [34] Haver
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János Sztrik 2001/08/05 [46] Koval
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János Sztrik 2001/08/05 [56] Scher
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János Sztrik 2001/08/05 [68] Sztri
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János Sztrik 2001/08/05 [79] Trive
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