## Heavy-traffic asymptotic expansions for the asymptotic decay rates (1994)

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Venue: | in the BMAP/G/1 queue. Stochastic Models |

Citations: | 15 - 10 self |

### BibTeX

@INPROCEEDINGS{Choudhury94heavy-trafficasymptotic,

author = {Gagan L. Choudhury and Ward Whitt},

title = {Heavy-traffic asymptotic expansions for the asymptotic decay rates},

booktitle = {in the BMAP/G/1 queue. Stochastic Models},

year = {1994},

pages = {453--498}

}

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### Abstract

versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, Perron-Frobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steady-state distributions in the BMAP / G /1 queue have asymptotically exponential tails. Here we develop asymptotic expansions for the asymptotic decay rates of these tail probabilities in powers of one minus the traffic intensity. The first term coincides with the decay rate of the exponential distribution arising in the standard heavy-traffic limit. The coefficients of these heavy-traffic expansions depend on the moments of the service-time distribution and the derivatives of the Perron-Frobenius eigenvalue δ(z) of the BMAP matrix generating function D(z) at z = 1. We give recursive formulas for the derivatives δ (k) ( 1). The asymptotic expansions provide the basis for efficiently computing the asymptotic decay rates as functions of the traffic intensity, i.e., the caudal characteristic curves. The asymptotic expansions also reveal what features of the model the asymptotic decay rates primarily depend upon. In particular, δ(z) coincides with the limiting time-average of the factorial cumulant generating function (the logarithm of the generating function) of the arrival counting process, and the derivatives δ (k) ( 1) coincide with the asymptotic factorial cumulants of the arrival counting process. This insight is important for admission control schemes in multi-service networks based in part on asymptotic decay rates. The interpretation helps identify appropriate statistics to compute from network traffic data in order to predict performance. 1.

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Citation Context ...derivatives. This can be used to calculate a k for arbitrary k. Second, Table 2 gives the first seven coefficients b k in (46) in terms of the coefficients a k in (45). Again the algorithm in Riordan =-=[32]-=- enables us to calculate b k for arbitrary k. Note that these formulas simplify in special cases. For example, if the service-time distribution is deterministic (D), then m k = 1 for all ks1. If the s... |

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Citation Context ... for stationary point processes, e.g., Daley and Vere-Jones [3, Exercise 10.4.7], together with a coupling argument to show that the initial nonstationarity is asymptotically negligible; see Lindvall =-=[24]-=-. Under appropriate regularity conditions, the statement in Theorem 2 can be improved to c k (t) = d (k) (1) t + g k + r k (t) (18) where r k (t) = o(1) as tsand sometimes even r k (t) = O(e - s k t )... |

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Citation Context ...rms of the derivatives d (k)sd (k) (1) and m k = EV k = f (k) (0). These are obtained from the composite function d(f(h)) plus (44). In Section 2.8 and Problem 32 on p. 47 of Riordan [29] and Riordan =-=[33]-=-, a recursive algorithm is given for the derivatives. This can be used to calculate a k for arbitrary k. Second, Table 2 gives the first seven coefficients b k in (46) in terms of the coefficients a k... |

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Citation Context ...rvice discipline and i.i.d. service times that are independent of a batch Markovian arrival process (BMAP). The BMAP is an alternative representation of the versatile Markovian point process of Neuts =-=[28, 30]-=- with an appealing simple notation, which was introduced by Lucantoni [25]. The BMAP/G/1 queue is equivalent to the N/G/1 queue considered by Ramaswami [31]. The BMAP generalizes the MAP by allowing b... |

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Citation Context ...e basic steady-state distributions in the BMAP/G/1 queue have asymptotically exponential tails. (For related work, see Asmussen [5], Asmussen and Perry [7], Baiocchi [8], Chang [9], Elwalid and Mitra =-=[14,15]-=-, Glynn and Whitt [18] and van Ommeren [40].) Our purpose here is to obtain heavy-traffic asymptotic expansions (in powers of 1 - r where r is the traffic intensity) for the asymptotic decay rates. Th... |

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Citation Context ...s a superposition of a large number of independent processes, the asymptotic decay rate alone often does not yield good approximations for tail probabilities. As in previous work on steadystate means =-=[16,20,39]-=-, more intricate approximations are evidently needed. Example 6.3 To illustrate how the results extend beyond BMAP arrival processes, we consider the G 0.5 /G 2 /1 queue, which has G 2sE 2 service tim... |

183 | Effective Bandwidths at Multi-class Queues
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Citation Context ... rest stems from the heavy-traffic expansion. For additional related work on admission control, see Chang [9], Elwalid and Mitra [14], Gibbens and Hunt [17], Guerin, Ahmadi and Naghshineh [19], Kelly =-=[23]-=-, Sohraby [37,38], Whitt [43] and references therein. In this context Sohraby [37,38] also considers (one-term) heavy-traffic approximations for the decay rates. It turns out that the first term of th... |

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Citation Context ...s a superposition of a large number of independent processes, the asymptotic decay rate alone often does not yield good approximations for tail probabilities. As in previous work on steadystate means =-=[16,20,39]-=-, more intricate approximations are evidently needed. Example 6.3 To illustrate how the results extend beyond BMAP arrival processes, we consider the G 0.5 /G 2 /1 queue, which has G 2sE 2 service tim... |

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Citation Context ... in great generality the basic steady-state distributions in the BMAP/G/1 queue have asymptotically exponential tails. (For related work, see Asmussen [5], Asmussen and Perry [7], Baiocchi [8], Chang =-=[9]-=-, Elwalid and Mitra [14,15], Glynn and Whitt [18] and van Ommeren [40].) Our purpose here is to obtain heavy-traffic asymptotic expansions (in powers of 1 - r where r is the traffic intensity) for the... |

149 | Logarithmic asymptotics for steady-state tail probabilities in a single-server queue
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Citation Context ...ributions in the BMAP/G/1 queue have asymptotically exponential tails. (For related work, see Asmussen [5], Asmussen and Perry [7], Baiocchi [8], Chang [9], Elwalid and Mitra [14,15], Glynn and Whitt =-=[18]-=- and van Ommeren [40].) Our purpose here is to obtain heavy-traffic asymptotic expansions (in powers of 1 - r where r is the traffic intensity) for the asymptotic decay rates. These asymptotic expansi... |

111 |
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Citation Context ...ovian arrival process (BMAP). The BMAP is an alternative representation of the versatile Markovian point process of Neuts [28, 30] with an appealing simple notation, which was introduced by Lucantoni =-=[25]-=-. The BMAP/G/1 queue is equivalent to the N/G/1 queue considered by Ramaswami [31]. The BMAP generalizes the MAP by allowing batch arrivals; the MAP generalizes the Markov modulated Poisson processes ... |

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Citation Context ...endent sources (see Theorem 3 in Section 2); the rest stems from the heavy-traffic expansion. For additional related work on admission control, see Chang [9], Elwalid and Mitra [14], Gibbens and Hunt =-=[17]-=-, Guerin, Ahmadi and Naghshineh [19], Kelly [23], Sohraby [37,38], Whitt [43] and references therein. In this context Sohraby [37,38] also considers (one-term) heavy-traffic approximations for the dec... |

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Citation Context ...affic expansion. For additional related work on admission control, see Chang [9], Elwalid and Mitra [14], Gibbens and Hunt [17], Guerin, Ahmadi and Naghshineh [19], Kelly [23], Sohraby [37,38], Whitt =-=[43]-=- and references therein. In this context Sohraby [37,38] also considers (one-term) heavy-traffic approximations for the decay rates. It turns out that the first term of the heavy-traffic expansion for... |

74 | Approximating a point process by a renewal process, I: Two basic methods - Whitt - 1982 |

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Citation Context ...rvice discipline and i.i.d. service times that are independent of a batch Markovian arrival process (BMAP). The BMAP is an alternative representation of the versatile Markovian point process of Neuts =-=[28, 30]-=- with an appealing simple notation, which was introduced by Lucantoni [25]. The BMAP/G/1 queue is equivalent to the N/G/1 queue considered by Ramaswami [31]. The BMAP generalizes the MAP by allowing b... |

41 |
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Citation Context ...e same type. Indeed, MAPs are sufficiently general that they can serve as approximations for any stationary point process (possibly at the expense of requiring large matrices); see Asmussen and Koole =-=[6]-=-. For an overview of the BMAP/G/1 queue, see Lucantoni [26]. In Abate, Choudhury and Whitt [2] we showed that in great generality the basic steady-state distributions in the BMAP/G/1 queue have asympt... |

39 | Exponential Approximation for Tail Probabilities in Queues
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Citation Context ... + d (2) (1), which corresponds to the asymptotic variance. Subsequent terms in the asymptotic expansion offer refinements to the basic heavytraffic approximation. As we found for the GI/G/1 queue in =-=[1]-=-, we find that a second term often provides a significant improvement, but that two terms is often a remarkably good approximation (for r not too small, e.g., rs0.6). The algorithm here provides a mea... |

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Citation Context ... h Vsfor other distributions. Let Q and L be the steady-state queue length (number in system) and workload (virtual waiting time) at an arbitrary time, which we assume are well defined. See Ramaswami =-=[31]-=-, Neuts [30], Lucantoni [25,26] and Abate, Choudhury and Whitt [2] for the transforms of the distributions of Q and L. We are interested in the asymptotic behavior of the tail probabilities. In great ... |

30 |
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(Show Context)
Citation Context ...om the heavy-traffic expansion. For additional related work on admission control, see Chang [9], Elwalid and Mitra [14], Gibbens and Hunt [17], Guerin, Ahmadi and Naghshineh [19], Kelly [23], Sohraby =-=[37,38]-=-, Whitt [43] and references therein. In this context Sohraby [37,38] also considers (one-term) heavy-traffic approximations for the decay rates. It turns out that the first term of the heavy-traffic e... |

29 |
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Citation Context ... In Abate, Choudhury and Whitt [2] we showed that in great generality the basic steady-state distributions in the BMAP/G/1 queue have asymptotically exponential tails. (For related work, see Asmussen =-=[5]-=-, Asmussen and Perry [7], Baiocchi [8], Chang [9], Elwalid and Mitra [14,15], Glynn and Whitt [18] and van Ommeren [40].) Our purpose here is to obtain heavy-traffic asymptotic expansions (in powers o... |

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Asymptotic formulas for Markov processes with applications to simulation
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Citation Context ...of the fundamental matrix of a CTMC instead of the fundamental matrix of a discrete-time Markov chain (DTMC). For additional discussion of fundamental matrices of CTMCs and more references, see Whitt =-=[42]-=-. Since DsD(1) is an infinitesimal generator of an irreducible CTMC, d(1) = 0, u(1) = p and v(1) = e. Let Y = (ep - D) - 1 and Z = Y - ep . (23) The matrix Z in (3.1) usually is called the fundamental... |

20 |
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Citation Context ...tain d (k) from the algorithm in Section 3. Given an explicit expression for the transform f(s)sEe sV , we can calculate any desired number of moments m k via the algorithm in Choudhury and Lucantoni =-=[10]-=-. Given either s or h (or approximations) and the derivatives, we can obtain an approximation for the other from (39)--(41), i.e., r h __ = d(s - 1 ) ~ ~ (s - 1 - 1) + . . . + d (k) k! (s - 1 - 1) k _... |

20 |
On the theory of general on-off sources with applications in high-speed networks
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Citation Context ...om the heavy-traffic expansion. For additional related work on admission control, see Chang [9], Elwalid and Mitra [14], Gibbens and Hunt [17], Guerin, Ahmadi and Naghshineh [19], Kelly [23], Sohraby =-=[37,38]-=-, Whitt [43] and references therein. In this context Sohraby [37,38] also considers (one-term) heavy-traffic approximations for the decay rates. It turns out that the first term of the heavy-traffic e... |

19 |
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Citation Context ...s. The asymptotic decay rates also can be computed by root finding, but the asymptotic expansions yield the asymptotic decay rates as functions of r (i.e., the caudal characteristic curves; see Neuts =-=[29]-=-), whereas the root finding must be repeated for each separate value of r. The asymptotic expansions also reveal what features of the model the asymptotic decay rates primarily depend upon. In particu... |

18 |
Measurements and approximations to describe the offered traffic and predict the average workload in a single-server queue
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Citation Context ..., while c s 2 ≡ m 2 − 1 is the squared coefficient of variation (variance divided by the square of the mean) of the service time. Hence, (49) agrees with the familiar heavy-traffic formula; e.g., see =-=[16]-=-. Paralleling [3], the two-term refined approximation here is ( 1 − ρ) η ∼ c 1 ( 1 − ρ) + c 2 2 2 _ _______ = _ 2 ________ ( 1 − ρ) ( 2 ) m 2 + δ + 3 (m 2 + δ ( 2 ) ) 3 2 ( 2 ) 2 2 ( 3m 2 + 3 (δ ) − 2... |

17 |
The BMAP/G/1 queue: A tutorial
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Citation Context ...ey can serve as approximations for any stationary point process (possibly at the expense of requiring large matrices); see Asmussen and Koole [6]. For an overview of the BMAP/G/1 queue, see Lucantoni =-=[26]-=-. In Abate, Choudhury and Whitt [2] we showed that in great generality the basic steady-state distributions in the BMAP/G/1 queue have asymptotically exponential tails. (For related work, see Asmussen... |

9 | Heavy-traffic expansions for the asymptotic decay rates in the BMAP/G/1 queue", Stochastic Models 10
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Citation Context ...iance and the asymptotic central third moment. Although our proofs depend on the BMAP structure, this characterization does not; it applies to arbitrary stochastic point processes. In Abate and Whitt =-=[3]-=- we showed that a heavy-traffic asymptotic expansion is possible for multi-channel queues in which the individual arrival and service channels are mutually independent renewal processes, and found the... |

9 |
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Citation Context ... Whitt [2] we showed that in great generality the basic steady-state distributions in the BMAP/G/1 queue have asymptotically exponential tails. (For related work, see Asmussen [5], Asmussen and Perry =-=[7]-=-, Baiocchi [8], Chang [9], Elwalid and Mitra [14,15], Glynn and Whitt [18] and van Ommeren [40].) Our purpose here is to obtain heavy-traffic asymptotic expansions (in powers of 1 - r where r is the t... |

9 |
Markovian arrival and service communication systems: spectral expansions, separability and Kronecker-product forms," submitted
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Citation Context ...e basic steady-state distributions in the BMAP/G/1 queue have asymptotically exponential tails. (For related work, see Asmussen [5], Asmussen and Perry [7], Baiocchi [8], Chang [9], Elwalid and Mitra =-=[14,15]-=-, Glynn and Whitt [18] and van Ommeren [40].) Our purpose here is to obtain heavy-traffic asymptotic expansions (in powers of 1 - r where r is the traffic intensity) for the asymptotic decay rates. Th... |

7 |
Measurements and Approximations to Describe Offered Traffic and Predict the Average Workload in a Single-Server Queue
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Citation Context ...ss, while c s 2sm 2 - 1 is the squared coefficient of variation (variance divided by the square of the mean) of the service time. Hence, (49) agrees with the familiar heavy-traffic formula; e.g., see =-=[16]-=-. Paralleling [3], the two-term refined approximation here is h ~ ~ c 1 (1 - r) + c 2 2 (1 - r) 2 ________ = m 2 + d (2) 2(1 - r) _________ + 3(m 2 + d (2) ) 3 2(3m 2 2 + 3(d (2) ) 2 - 2m 3 - 2d (3) )... |

7 |
Neuts, The first two moments matrices of the counts for the Markovian arrival process, Stochastic Models
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Citation Context ... expressions for the second-order terms g k in (18) may be useful for developing refined approximations. For the first two cumulants c k (t), explicit expressions are also given in Narayana and Neuts =-=[27]-=- and Chapter 5 of Neuts [30]. The superposition of n independent BMAPs can be represented as another BMAP with an auxiliary phase state space equal to the product of the n individual auxiliary phase s... |

5 |
Squeezing the most out of ATM. Submitted
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Citation Context ...r percentiles; see Example 6.2 below. However, we have found that the asymptotic constant can be very far from 1 when the arrival process is the superposition of a large number of independent sources =-=[12]-=-. In such circumstances, we evidently need more than the asymptotic decay rate to find good approximations for tail probabilities. 2. The Batch Markovian Arrival Process In this section we review the ... |

4 |
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Citation Context ...mpare approximations with exact values for the percentiles of the steady-state waiting time when there are one and two component streams. The exact percentile values are computed using the program in =-=[11]-=-. (Bisection search is used to find the percentiles with the algorithm for the tail probabilities.) For the approximations, we always use the exact value of the asymptotic decay rate h (which does not... |

3 |
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Citation Context ...showed that in great generality the basic steady-state distributions in the BMAP/G/1 queue have asymptotically exponential tails. (For related work, see Asmussen [5], Asmussen and Perry [7], Baiocchi =-=[8]-=-, Chang [9], Elwalid and Mitra [14,15], Glynn and Whitt [18] and van Ommeren [40].) Our purpose here is to obtain heavy-traffic asymptotic expansions (in powers of 1 - r where r is the traffic intensi... |

3 |
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Citation Context ...tion 2); the rest stems from the heavy-traffic expansion. For additional related work on admission control, see Chang [9], Elwalid and Mitra [14], Gibbens and Hunt [17], Guerin, Ahmadi and Naghshineh =-=[19]-=-, Kelly [23], Sohraby [37,38], Whitt [43] and references therein. In this context Sohraby [37,38] also considers (one-term) heavy-traffic approximations for the decay rates. It turns out that the firs... |

2 |
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(Show Context)
Citation Context ...Theorem 2 can be improved to c k (t) = d (k) (1) t + g k + r k (t) (18) where r k (t) = o(1) as tsand sometimes even r k (t) = O(e - s k t ) as tswhere s k is a positive constant. In particular Smith =-=[35,36]-=- obtained such results for renewal processes and cumulative processes (associated with regenerative processes). Note that here N(t) is indeed a cumulative process; as regeneration times we can take su... |

2 |
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Citation Context .../G/1 queue have asymptotically exponential tails. (For related work, see Asmussen [5], Asmussen and Perry [7], Baiocchi [8], Chang [9], Elwalid and Mitra [14,15], Glynn and Whitt [18] and van Ommeren =-=[40]-=-.) Our purpose here is to obtain heavy-traffic asymptotic expansions (in powers of 1 - r where r is the traffic intensity) for the asymptotic decay rates. These asymptotic expansions provide a conveni... |

1 |
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Citation Context ...Theorem 2 can be improved to c k (t) = d (k) (1) t + g k + r k (t) (18) where r k (t) = o(1) as tsand sometimes even r k (t) = O(e - s k t ) as tswhere s k is a positive constant. In particular Smith =-=[35,36]-=- obtained such results for renewal processes and cumulative processes (associated with regenerative processes). Note that here N(t) is indeed a cumulative process; as regeneration times we can take su... |

1 |
The BMAP / G /1 queue: a tutorial
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- 1993
(Show Context)
Citation Context ...can serve as approximations for any stationary point process (possibly at the expense of requiring large matrices); see Asmussen and Koole [6]. For an overview of the BMAP / G /1 queue, see Lucantoni =-=[26]-=-. In Abate, Choudhury and Whitt [2] we showed that in great generality the basic steady-state distributions in the BMAP / G /1 queue havesasymptotically exponential tails. (For related work, see Asmus... |