Journal articles: 'Heavy-tailed workloads'

1

Psounis, Konstantinos, Pablo Molinero-Fernández, Balaji Prabhakar, and Fragkiskos Papadopoulos. "Systems with multiple servers under heavy-tailed workloads." Performance Evaluation 62, no. 1-4 (October 2005): 456–74. http://dx.doi.org/10.1016/j.peva.2005.07.030.

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Tai, Jianzhe, Zhen Li, Jiahui Chen, and Ningfang Mi. "Load balancing for cluster systems under heavy-tailed and temporal dependent workloads." Simulation Modelling Practice and Theory 44 (May 2014): 63–77. http://dx.doi.org/10.1016/j.simpat.2014.03.006.

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Foss, Sergey, and Masakiyo Miyazawa. "Two-node fluid network with a heavy-tailed random input: the strong stability case." Journal of Applied Probability 51, A (December 2014): 249–65. http://dx.doi.org/10.1017/s0021900200021318.

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We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

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Foss, Sergey, and Masakiyo Miyazawa. "Two-node fluid network with a heavy-tailed random input: the strong stability case." Journal of Applied Probability 51, A (December 2014): 249–65. http://dx.doi.org/10.1239/jap/1417528479.

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We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

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Janevski, Nikola, and Katerina Goseva-Popstojanova. "Session Reliability of Web Systems under Heavy-Tailed Workloads: An Approach Based on Design and Analysis of Experiments." IEEE Transactions on Software Engineering 39, no. 8 (August 2013): 1157–78. http://dx.doi.org/10.1109/tse.2013.3.

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Tang, Cheng-Jen, and Miau-Ru Dai. "Modeling and Analysis of Queueing-Based Vary-On/Vary-Off Schemes for Server Clusters." Mathematical Problems in Engineering 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/594264.

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A cloud system usually consists of a lot of server clusters handling various applications. To satisfy the increasing demands, especially for the front-end web applications, the computing capacity of a cloud system is often allocated for the peak demand. Such installation causes resource underutilization during the off-peak hours. Vary-On/Vary-Off (VOVO) schemes concentrate workloads on some servers instead of distributing them across all servers in a cluster to reduce idle energy waste. Recent VOVO schemes adopt queueing theory to model the arrival process and the service process for determining the number of powered-on servers. For the arrival process, Poisson process can be safely assumed in web services due to the large number of independent sources. On the other hand, the heavy-tailed distribution of service times is observed in real web systems. However, there are no exact solutions to determine the performance forM/heavy-tailed/mqueues. Therefore, this paper presents two queueing-based sizing approximations for Poisson and non-Poisson governed service processes. The simulation results of the proposed approximations are analyzed and evaluated by comparing with the simulated system running at full capacity. This relative measurement indicates that the Pareto distributed service process may be adequately modeled by memoryless queues when VOVO schemes are adopted.

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Borst, Sem, Michel Mandjes, and Miranda van Uitert. "Generalized processor sharing queues with heterogeneous traffic classes." Advances in Applied Probability 35, no. 03 (September 2003): 806–45. http://dx.doi.org/10.1017/s0001867800012556.

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We consider a queue fed by a mixture of light-tailed and heavy-tailed traffic. The two traffic flows are served in accordance with the generalized processor sharing (GPS) discipline. GPS-based scheduling algorithms, such as weighted fair queueing (WFQ), have emerged as an important mechanism for achieving service differentiation in integrated networks. We derive the asymptotic workload behaviour of the light-tailed traffic flow under the assumption that its GPS weight is larger than its traffic intensity. The GPS mechanism ensures that the workload is bounded above by that in an isolated system with the light-tailed flow served in isolation at a constant rate equal to its GPS weight. We show that the workload distribution is, in fact, asymptotically equivalent to that in the isolated system, multiplied by a certain prefactor, which accounts for the interaction with the heavy-tailed flow. Specifically, the prefactor represents the probability that the heavy-tailed flow is backlogged long enough for the light-tailed flow to reach overflow. The results provide crucial qualitative insight in the typical overflow scenario.

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Borst, Sem, Michel Mandjes, and Miranda van Uitert. "Generalized processor sharing queues with heterogeneous traffic classes." Advances in Applied Probability 35, no. 3 (September 2003): 806–45. http://dx.doi.org/10.1239/aap/1059486830.

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We consider a queue fed by a mixture of light-tailed and heavy-tailed traffic. The two traffic flows are served in accordance with the generalized processor sharing (GPS) discipline. GPS-based scheduling algorithms, such as weighted fair queueing (WFQ), have emerged as an important mechanism for achieving service differentiation in integrated networks. We derive the asymptotic workload behaviour of the light-tailed traffic flow under the assumption that its GPS weight is larger than its traffic intensity. The GPS mechanism ensures that the workload is bounded above by that in an isolated system with the light-tailed flow served in isolation at a constant rate equal to its GPS weight. We show that the workload distribution is, in fact, asymptotically equivalent to that in the isolated system, multiplied by a certain prefactor, which accounts for the interaction with the heavy-tailed flow. Specifically, the prefactor represents the probability that the heavy-tailed flow is backlogged long enough for the light-tailed flow to reach overflow. The results provide crucial qualitative insight in the typical overflow scenario.

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9

Borst, Sem, and Bert Zwart. "A reduced-peak equivalence for queues with a mixture of light-tailed and heavy-tailed input flows." Advances in Applied Probability 35, no. 03 (September 2003): 793–805. http://dx.doi.org/10.1017/s0001867800012544.

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We determine the exact large-buffer asymptotics for a mixture of light-tailed and heavy-tailed input flows. Earlier studies have found a ‘reduced-load equivalence’ in situations where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is larger than the service rate. In that case, the workload is asymptotically equivalent to that in a reduced system, which consists of a certain ‘dominant’ subset of the heavy-tailed flows, with the service rate reduced by the mean rate of all other flows. In the present paper, we focus on the opposite case where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is smaller than the service rate. Under mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a somewhat ‘dual’ reduced system, multiplied by a certain prefactor. The reduced system now consists of only the light-tailed flows, with the service rate reduced by the peak rate of the heavy-tailed flows. The prefactor represents the probability that the heavy-tailed flows have sent at their peak rate for more than a certain amount of time, which may be interpreted as the ‘time to overflow’ for the light-tailed flows in the reduced system. The results provide crucial insight into the typical overflow scenario.

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10

Borst, Sem, and Bert Zwart. "A reduced-peak equivalence for queues with a mixture of light-tailed and heavy-tailed input flows." Advances in Applied Probability 35, no. 3 (September 2003): 793–805. http://dx.doi.org/10.1239/aap/1059486829.

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We determine the exact large-buffer asymptotics for a mixture of light-tailed and heavy-tailed input flows. Earlier studies have found a ‘reduced-load equivalence’ in situations where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is larger than the service rate. In that case, the workload is asymptotically equivalent to that in a reduced system, which consists of a certain ‘dominant’ subset of the heavy-tailed flows, with the service rate reduced by the mean rate of all other flows. In the present paper, we focus on the opposite case where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is smaller than the service rate. Under mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a somewhat ‘dual’ reduced system, multiplied by a certain prefactor. The reduced system now consists of only the light-tailed flows, with the service rate reduced by the peak rate of the heavy-tailed flows. The prefactor represents the probability that the heavy-tailed flows have sent at their peak rate for more than a certain amount of time, which may be interpreted as the ‘time to overflow’ for the light-tailed flows in the reduced system. The results provide crucial insight into the typical overflow scenario.

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11

Resnick, Sidney, and Gennady Samorodnitsky. "A Heavy Traffic Approximation for Workload Processes with Heavy Tailed Service Requirements." Management Science 46, no. 9 (September 2000): 1236–48. http://dx.doi.org/10.1287/mnsc.46.9.1236.12234.

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12

Leipus, Remigijus, and Donatas Surgailis. "On Long-Range Dependence in Regenerative Processes Based on a General ON/OFF Scheme." Journal of Applied Probability 44, no. 02 (June 2007): 379–92. http://dx.doi.org/10.1017/s002190020000303x.

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In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

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Leipus, Remigijus, and Donatas Surgailis. "On Long-Range Dependence in Regenerative Processes Based on a General ON/OFF Scheme." Journal of Applied Probability 44, no. 02 (June 2007): 379–92. http://dx.doi.org/10.1017/s0021900200117899.

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In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

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Leipus, Remigijus, and Donatas Surgailis. "On Long-Range Dependence in Regenerative Processes Based on a General ON/OFF Scheme." Journal of Applied Probability 44, no. 2 (June 2007): 379–92. http://dx.doi.org/10.1239/jap/1183667408.

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In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

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15

Es-Saghouani, Abdelghafour, and Michel Mandjes. "On the Correlation Structure of a Lévy-Driven Queue." Journal of Applied Probability 45, no. 4 (December 2008): 940–52. http://dx.doi.org/10.1239/jap/1231340225.

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In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Qt)t≥0, with a focus on the correlation structure. With the correlation function defined asr(t) := cov(Q0,Qt) / var(Q0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0∞r(t)e-ϑtdt. This expression allows us to prove thatr(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show thatr(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics ofr(t), for larget, for the cases of light-tailed and heavy-tailed Lévy inputs.

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16

Es-Saghouani, Abdelghafour, and Michel Mandjes. "On the Correlation Structure of a Lévy-Driven Queue." Journal of Applied Probability 45, no. 04 (December 2008): 940–52. http://dx.doi.org/10.1017/s0021900200004897.

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In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Q t ) t≥0, with a focus on the correlation structure. With the correlation function defined as r(t) := cov(Q 0, Q t ) / var(Q 0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0 ∞ r(t)e-ϑt dt. This expression allows us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that r(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics of r(t), for large t, for the cases of light-tailed and heavy-tailed Lévy inputs.

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17

Mees, R., and R. Chase. "Relating Burning Index to Wildfire Workload Over Broad Geographic Areas." International Journal of Wildland Fire 1, no. 4 (1991): 235. http://dx.doi.org/10.1071/wf9910235.

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The burning index of the National Fire Danger Rating System is designed to measure potential fire workload over broad geographic areas that can be repre sented as being homogeneous with respect to fuel, topo graphic, and weather conditions. The utility of this index is confirmed by its relation to three measures of fire workload-number of fires, area burned, and number of personnel used in fire suppression for National Forests in southern California. The distributions of these mea sures over 15 years were skewed heavily to the right ("heavy-tailed distributions"). We selected the75 th, 90th, and 95th percentile values of each distribution at ten percentile values of the burning index to investigate and display the association between fire workload and the burning index. The results provide a distinct view of the direct relationship between wildfire workload and critical burning index values for the southern California area as a whole, and point to the potential value of this approach for anticipating fire control problems in other areas.

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18

Lee, Chihoon. "On the Return Time for a Reflected Fractional Brownian Motion Process on the Positive Orthant." Journal of Applied Probability 48, no. 1 (March 2011): 145–53. http://dx.doi.org/10.1239/jap/1300198141.

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We consider a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R+d, with drift r0 ∈ Rd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a return time result for the RFBM process Z; that is, for some δ, κ > 0, supx∈BEx[τB(δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τB(δ) = inf{t ≥ δ : Z(t) ∈ B}. Similar results are known for reflected processes driven by standard Brownian motions, and our result can be viewed as their FBM counterpart. Our motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

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Lee, Chihoon. "On the Return Time for a Reflected Fractional Brownian Motion Process on the Positive Orthant." Journal of Applied Probability 48, no. 01 (March 2011): 145–53. http://dx.doi.org/10.1017/s0021900200007683.

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We consider a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R + d , with drift r 0 ∈ R d and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a return time result for the RFBM process Z; that is, for some δ, κ > 0, sup x∈B E x [τ B (δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τ B (δ) = inf{t ≥ δ : Z(t) ∈ B}. Similar results are known for reflected processes driven by standard Brownian motions, and our result can be viewed as their FBM counterpart. Our motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

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20

Vatamidou, Eleni, Ivo Adan, Maria Vlasiou, and Bert Zwart. "Corrected phase-type approximations for the workload of the MAP/G/1 queue with heavy-tailed service times." ACM SIGMETRICS Performance Evaluation Review 41, no. 2 (August 27, 2013): 53–55. http://dx.doi.org/10.1145/2518025.2518036.

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21

Budhiraja, Amarjit, Vladas Pipiras, and Xiaoming Song. "Admission Control for Multidimensional Workload input with Heavy Tails and Fractional Ornstein-Uhlenbeck Process." Advances in Applied Probability 47, no. 2 (June 2015): 476–505. http://dx.doi.org/10.1239/aap/1435236984.

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The infinite source Poisson arrival model with heavy-tailed workload distributions has attracted much attention, especially in the modeling of data packet traffic in communication networks. In particular, it is well known that under suitable assumptions on the source arrival rate, the centered and scaled cumulative workload input process for the underlying processing system can be approximated by fractional Brownian motion. In many applications one is interested in the stabilization of the work inflow to the system by modifying the net input rate, using an appropriate admission control policy. In this paper we study a natural family of admission control policies which keep the associated scaled cumulative workload input asymptotically close to a prespecified linear trajectory, uniformly over time. Under such admission control policies and with natural assumptions on arrival distributions, suitably scaled and centered cumulative workload input processes are shown to converge weakly in the path space to the solution of a d-dimensional stochastic differential equation driven by a Gaussian process. It is shown that the admission control policy achieves moment stabilization in that the second moment of the solution to the stochastic differential equation (averaged over the d-stations) is bounded uniformly for all times. In one special case of control policies, as time approaches ∞, we obtain a fractional version of a stationary Ornstein-Uhlenbeck process that is driven by fractional Brownian motion with Hurst parameter H > ½.

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Budhiraja, Amarjit, Vladas Pipiras, and Xiaoming Song. "Admission Control for Multidimensional Workload input with Heavy Tails and Fractional Ornstein-Uhlenbeck Process." Advances in Applied Probability 47, no. 02 (June 2015): 476–505. http://dx.doi.org/10.1017/s0001867800007941.

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The infinite source Poisson arrival model with heavy-tailed workload distributions has attracted much attention, especially in the modeling of data packet traffic in communication networks. In particular, it is well known that under suitable assumptions on the source arrival rate, the centered and scaled cumulative workload input process for the underlying processing system can be approximated by fractional Brownian motion. In many applications one is interested in the stabilization of the work inflow to the system by modifying the net input rate, using an appropriate admission control policy. In this paper we study a natural family of admission control policies which keep the associated scaled cumulative workload input asymptotically close to a prespecified linear trajectory, uniformly over time. Under such admission control policies and with natural assumptions on arrival distributions, suitably scaled and centered cumulative workload input processes are shown to converge weakly in the path space to the solution of a d-dimensional stochastic differential equation driven by a Gaussian process. It is shown that the admission control policy achieves moment stabilization in that the second moment of the solution to the stochastic differential equation (averaged over the d-stations) is bounded uniformly for all times. In one special case of control policies, as time approaches ∞, we obtain a fractional version of a stationary Ornstein-Uhlenbeck process that is driven by fractional Brownian motion with Hurst parameter H > ½.

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23

Dębicki, Krzysztof, Abdelghafour Es-Saghouani, and Michel Mandjes. "Transient Asymptotics of Lévy-Driven Queues." Journal of Applied Probability 47, no. 1 (March 2010): 109–29. http://dx.doi.org/10.1239/jap/1269610820.

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With (Qt)t denoting the stationary workload process in a queue fed by a Lévy input process (Xt)t, this paper focuses on the asymptotics of rare event probabilities of the type P(Q0 > pB, QTB > qB) for given positive numbers p and q, and a positive deterministic function TB. We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to TB being sublinear (i.e. TB/B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that TB is superlinear (i.e. TB / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to TB / B2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for TB increasing superlinearly but subquadratically. We pay special attention to the case TB = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

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24

Lee, Chihoon. "A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant." Journal of Applied Probability 48, no. 3 (September 2011): 820–31. http://dx.doi.org/10.1239/jap/1316796917.

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We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+d, with drift r0 ∈ ℝd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= Ex[V(Ž̆(1))] − V(x) ≤ −βV(x) + b1C(x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

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25

Dębicki, Krzysztof, Abdelghafour Es-Saghouani, and Michel Mandjes. "Transient Asymptotics of Lévy-Driven Queues." Journal of Applied Probability 47, no. 01 (March 2010): 109–29. http://dx.doi.org/10.1017/s0021900200006434.

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With (Q t ) t denoting the stationary workload process in a queue fed by a Lévy input process (X t ) t , this paper focuses on the asymptotics of rare event probabilities of the type P(Q 0 > pB, Q T B > qB) for given positive numbers p and q, and a positive deterministic function T B . We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to T B being sublinear (i.e. T B /B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that T B is superlinear (i.e. T B / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to T B / B 2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for T B increasing superlinearly but subquadratically. We pay special attention to the case T B = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

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26

Lee, Chihoon. "A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant." Journal of Applied Probability 48, no. 03 (September 2011): 820–31. http://dx.doi.org/10.1017/s0021900200008342.

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We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+ d , with drift r 0 ∈ ℝ d and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= E x [V(Ž̆(1))] − V(x) ≤ −βV(x) + b 1 C (x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

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27

Behme, Anita, and Philipp Lukas Strietzel. "A $$2~{\times }~2$$ random switching model and its dual risk model." Queueing Systems, April 5, 2021. http://dx.doi.org/10.1007/s11134-021-09697-9.

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AbstractIn this article, a special case of two coupled M/G/1-queues is considered, where two servers are exposed to two types of jobs that are distributed among the servers via a random switch. In this model, the asymptotic behavior of the workload buffer exceedance probabilities for the two single servers/both servers together/one (unspecified) server is determined. Hereby, one has to distinguish between jobs that are either heavy-tailed or light-tailed. The results are derived via the dual risk model of the studied coupled M/G/1-queues for which the asymptotic behavior of different ruin probabilities is determined.

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Journal articles on the topic 'Heavy-tailed workloads'

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