SET-VALUED PERFORMANCE APPROXIMATIONS FOR THE QUEUE GIVEN PARTIAL INFORMATION

2020 ◽  
pp. 1-23
Author(s):  
Yan Chen ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Decay Rate ◽  
Waiting Time ◽  
Partial Information ◽  
Upper And Lower Bounds ◽  
Interarrival Time ◽  
Limited Information ◽  
Wide Range ◽  
The Mean ◽  
Third Moment

In order to understand queueing performance given only partial information about the model, we propose determining intervals of likely values of performance measures given that limited information. We illustrate this approach for the mean steady-state waiting time in the $GI/GI/K$ queue. We start by specifying the first two moments of the interarrival-time and service-time distributions, and then consider additional information about these underlying distributions, in particular, a third moment and a Laplace transform value. As a theoretical basis, we apply extremal models yielding tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability. We illustrate by constructing the theoretically justified intervals of values for the decay rate and the associated heuristically determined interval of values for the mean waiting times. Without extra information, the extremal models involve two-point distributions, which yield a wide range for the mean. Adding constraints on the third moment and a transform value produces three-point extremal distributions, which significantly reduce the range, producing practical levels of accuracy.

scholarly journals Processor-sharing and random-service queues with semi-Markovian arrivals

2005 ◽  
Vol 42 (02) ◽  
pp. 478-490
Author(s):  
De-An Wu ◽  
Hideaki Takagi
Keyword(s):  
Waiting Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Interarrival Time ◽  
Size Distributions ◽  
Processor Sharing ◽  
Queue Size ◽  
Mean Waiting Time ◽  
The Mean

We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-lcustomer who, upon his arrival, meetskcustomers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-lcustomer who, upon his arrival, meetsk+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

scholarly journals Statistical energy conservation principle for inhomogeneous turbulent dynamical systems

2015 ◽  
Vol 112 (29) ◽  
pp. 8937-8941 ◽  
Author(s):  
Andrew J. Majda
Keyword(s):  
Dynamical Systems ◽  
Energy Conservation ◽  
Upper And Lower Bounds ◽  
Grand Challenge ◽  
Mean Flow ◽  
Turbulent Fluctuations ◽  
Conservation Principle ◽  
Wide Range ◽  
The Mean ◽  
Energy Conservation Principle

Understanding the complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence as well as climate atmosphere ocean science is a grand challenge of contemporary science with important societal impact. In such inhomogeneous turbulent dynamical systems there is a large dimensional phase space with a large dimension of unstable directions where a large-scale ensemble mean and the turbulent fluctuations exchange energy and strongly influence each other. These complex features strongly impact practical prediction and uncertainty quantification. A systematic energy conservation principle is developed here in a Theorem that precisely accounts for the statistical energy exchange between the mean flow and the related turbulent fluctuations. This statistical energy is a sum of the energy in the mean and the trace of the covariance of the fluctuating turbulence. This result applies to general inhomogeneous turbulent dynamical systems including the above applications. The Theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self-contained treatment is presented below. Corollary 1 and Corollary 2 illustrate the power of the method with general closed differential equalities for the statistical energy in time either exactly or with upper and lower bounds, provided that the negative symmetric dissipation matrix is diagonal in a suitable basis. Implications of the energy principle for low-order closure modeling and automatic estimates for the single point variance are discussed below.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (02) ◽  
pp. 461-466
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

scholarly journals Adapting Footfall Rhythmicity to Auditory Perturbations Affects Resilience of Locomotor Behavior: A Proof-of-Concept Study

2021 ◽  
Vol 15 ◽  
Author(s):  
Deepak K. Ravi ◽  
Caroline C. Heimhofer ◽  
William R. Taylor ◽  
Navrag B. Singh
Keyword(s):  
Steady State ◽  
Fall Risk ◽  
Recovery Time ◽  
Center Of Mass ◽  
Locomotor Behavior ◽  
Proof Of Concept ◽  
Limited Information ◽  
Novel Approach ◽  
Temporal Adaptation ◽  
The Mean

For humans, the ability to effectively adapt footfall rhythm to perturbations is critical for stable locomotion. However, only limited information exists regarding how dynamic stability changes when individuals modify their footfall rhythm. In this study, we recorded 3D kinematic activity from 20 participants (13 males, 18–30 years old) during walking on a treadmill while synchronizing with an auditory metronome sequence individualized to their baseline walking characteristics. The sequence then included unexpected temporal perturbations in the beat intervals with the subjects required to adapt their footfall rhythm accordingly. Building on a novel approach to quantify resilience of locomotor behavior, this study found that, in response to auditory perturbation, the mean center of mass (COM) recovery time across all participants who showed deviation from steady state (N = 15) was 7.4 (8.9) s. Importantly, recovery of footfall synchronization with the metronome beats after perturbation was achieved prior (+3.4 [95.0% CI +0.1, +9.5] s) to the recovery of COM kinematics. These results highlight the scale of temporal adaptation to perturbations and provide implications for understanding regulation of rhythm and balance. Thus, our study extends the sensorimotor synchronization paradigm to include analysis of COM recovery time toward improving our understanding of an individual’s resilience to perturbations and potentially also their fall risk.

scholarly journals On Mx/(G1G2)/1/G(BS)/Vs vacation queue with two types of general heterogeneous service

2005 ◽  
Vol 2005 (3) ◽  
pp. 123-135 ◽  
Author(s):  
Kailash C. Madan ◽  
Z. R. Al-Rawi ◽  
Amjad D. Al-Nasser
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Waiting Time ◽  
Busy Period ◽  
Single Server ◽  
Queue Size ◽  
Single Vacation ◽  
The Mean ◽  
Expected Waiting Time ◽  
Heterogeneous Service

We analyze a batch arrival queue with a single server providing two kinds of general heterogeneous service. Just before his service starts, a customer may choose one of the services and as soon as a service (of any kind) gets completed, the server may take a vacation or may continue staying in the system. The vacation times are assumed to be general and the server vacations are based on Bernoulli schedules under a single vacation policy. We obtain explicit queue size distribution at a random epoch as well as at a departure epoch and also the mean busy period of the server under the steady state. In addition, some important performance measures such as the expected queue size and the expected waiting time of a customer are obtained. Further, some interesting particular cases are also discussed.

Sustained, synchronous oscillations in discharge of sustaining fibers of crayfish optic nerve

1976 ◽  
Vol 39 (6) ◽  
pp. 1257-1271 ◽  
Author(s):  
R. M. Glantz ◽  
H. B. Nudelman
Keyword(s):  
Steady State ◽  
Time Course ◽  
Cycle Phase ◽  
Cycle Period ◽  
Uniform Field ◽  
Excitatory Pathways ◽  
Wide Range ◽  
Cross Correlations ◽  
The Common ◽  
The Mean

1. The regularity of the sustaining fiber (SF) steady-state discharge increases with the intensity of a uniform field of illumination. 2. In a high levels of illumination SFs exhibit a highly periodic or rhythmically bursting steady-state discharge. 3. The period of the burst cycle is independent of the light intensity and the mean firing rate. 4. Multimodal interspike-interval histograms in which successive modes decline exponentially suggest that the discharge arises from a regular input. 5. In conditions of uniform illumination, the regular input must be common to all or several SFs since the period of the burst cycle of simultaneously monitored SFs are the same and cross-correlations indicate that the SFs burst in approximate synchrony. 6. It is proposed that the common, regular input is a network of presynaptic elements exhibiting recurrent lateral inhibition. Evidence supporting this hypothesis is that following a brief light pulse to the inhibitory surround, SFs exhibit cycles of silence and bursting, phase-locked to the stimulus but with a period equal to the steady-state burst-cycle period. 7. Furthermore, lateral inhibitory influences are common to several or all SFs as indicated by the close similarity in response time course of two SFs subjected to a common inhibitory stimulus. 8. The response of the lateral pathways to sinusoidally modulated illumination is highly selective to frequencies near the burst-repetition rate. The excitatory pathways exhibit frequency modulation or entrainment over a wide range of input frequencies. 9. The intersustaining burst cycle phase is under partial visual stimulus control (position and intensity) and the possibility of phase coding in the SF ensemble is discussed.

Relations for the workload of the GI/G/s queue

1985 ◽  
Vol 17 (4) ◽  
pp. 887-904 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Mean Waiting Time ◽  
Special Cases ◽  
The Mean ◽  
Total Workload

A relation between the distributions of the workload of an arbitrary server and the waiting time is obtained for a GI/G/s queue in steady state. An expression for the total workload of the system (total unprocessed work) is also found. Several well-known results emerge as special cases. Various useful relations, for instance bounds and approximations for the mean waiting time, are also given.

scholarly journals Processor-sharing and random-service queues with semi-Markovian arrivals

2005 ◽  
Vol 42 (2) ◽  
pp. 478-490 ◽  
Author(s):  
De-An Wu ◽  
Hideaki Takagi
Keyword(s):  
Waiting Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Interarrival Time ◽  
Size Distributions ◽  
Processor Sharing ◽  
Queue Size ◽  
Mean Waiting Time ◽  
The Mean

We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-l customer who, upon his arrival, meets k customers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-l customer who, upon his arrival, meets k+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

Relations for the workload of the GI/G/s queue

1985 ◽  
Vol 17 (04) ◽  
pp. 887-904
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Mean Waiting Time ◽  
Special Cases ◽  
The Mean ◽  
Total Workload

A relation between the distributions of the workload of an arbitrary server and the waiting time is obtained for a GI/G/s queue in steady state. An expression for the total workload of the system (total unprocessed work) is also found. Several well-known results emerge as special cases. Various useful relations, for instance bounds and approximations for the mean waiting time, are also given.

scholarly journals Analytical Solution of Safety Stock Determination in Case of Uncertain Unimodal Lead-Time Demand

2018 ◽  
Vol 21 ◽  
pp. 75-80
Author(s):  
Gerrit Karel Janssens ◽  
Lotte Verdonck ◽  
Katrien Ramaekers
Keyword(s):  
Lead Time ◽  
Partial Information ◽  
Service Level ◽  
Performance Measure ◽  
Service Performance ◽  
Upper And Lower Bounds ◽  
Expected Number ◽  
Safety Stock ◽  
A Value ◽  
The Mean

As companies state that a delivery service is important to their customers, an out-of-stock is considered harmful and therefore they keep safety stock in case of uncertain demand. For decision making on the level of safety stock a complete formulation of the distributional form of the demand during lead time is required. In practice, this information may not be available. In such a case, only partial information on the distribution might be available, such as the range, the mode, the mean or the variance. Given a value for a service performance measure, the decision maker, in this case, is not confronted with a single value for the safety stock but rather with an interval. The present research shows how upper and lower bounds of the safety stock are obtained in an analytical way, given a pre-specified service level using a service performance measure, called ‘expected number of units short’. The technique is also illustrated and compared within the framework of the research.

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