Mean Sojourn Time Research Articles

Confined random motion with Laplace and Linnik statistics

In this paper we reveal that the conjugate property of Bernstein functions connects the tempered subdiffusion with the confinement. The interpretation of anomalous diffusion tending to the confinement is that diffusive motion, accompanied by multiple-trapping events with infinite mean sojourn time, is transformed into pure jumps, restricted in confined environment. This model, just like the tempered one, is generally valid in a wide class of problems of transport in random media that include live cells, relaxation in heterogeneous substances, and jump-diffusion. A relation of our approach to the popular diffusive diffusivity is explained.

Inference of Sojourn Time and Transition Density using the NLST X-ray Screening Data in Lung Cancer.

The aim of this study is to provide statistical inference of the sojourn time and the transition probability from the disease free to the preclinical state of lung cancer for male and female smokers using lung cancer data from the National Lung Screening Trial (NLST). We applied a likelihood function to the lung cancer data, to obtain Bayesian inference of the transition probability and the sojourn time distribution. A log-normal distribution was used for the transition probability density function multiplied by 30%, and a Weibull distribution was used to model the sojourn time in the preclinical state. The estimate of screening sensitivity is 0.61 for males and 0.62 for females. Early transition happened before age 50 and lasted until after age 90. The transition probability from the disease free to the preclinical state has a single maximum at around age 73 for males and 72 for females. For male, the Bayesian posterior mean, and median sojourn time are 1.33 and 1.27 years, respectively. For female, the corresponding posterior mean, and median sojourn time are 1.23 and 1.21 years, respectively. Our estimation showed that male smokers are more vulnerable to lung cancer, because they have a higher transition probability density than the same aged female smokers. The female smokers have a slightly shorter mean sojourn time than the male, meaning that they are quicker to develop clinical symptom of lung cancer.

Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination

<p style='text-indent:20px;'>In this paper, a stochastic SIRS epidemic model with nonlinear incidence and vaccination is formulated to investigate the transmission dynamics of infectious diseases. The model not only incorporates the white noise but also the external environmental noise which is described by semi-Markov process. We first derive the explicit expression for the basic reproduction number of the model. Then the global dynamics of the system is studied in terms of the basic reproduction number and the intensity of white noise, and sufficient conditions for the extinction and persistence of the disease are both provided. Furthermore, we explore the sensitivity analysis of <inline-formula><tex-math id="M1">\begin{document}$ R_0^s $\end{document}</tex-math></inline-formula> with each semi-Markov switching under different distribution functions. The results show that the dynamics of the entire system is not related to its switching law, but has a positive correlation to its mean sojourn time in each subsystem. The basic reproduction number we obtained in the paper can be applied to all piecewise-stochastic semi-Markov processes, and the results of the sensitivity analysis can be regarded as a prior work for optimal control.</p>

Modeling and performance analysis of smart map application in the Multi-access Edge Computing paradigm

Multi-access Edge Computing (MEC) recognized as an emerging technology that provides cloud computing services in the proximity of Radio Access Network (RAN). Mobile subscribers could offload their computation-intensive or memory-intensive tasks to the nearest Base Stations (BS) and deliver the cloud services from MEC servers. This technology can meet the requirements of low latency and location awareness. Smart map applications are an indispensable part of our digital life in large cities. In this paper, we introduce how the map application could gain from MEC technology. We concentrate on performance modeling and analysis of direction-finding in the map application. We study the impact of variation of workload, the number of servers, queue length, and connection failure on the task rejection probability and mean sojourn time. We model each phase of direction-finding service by the Markov Reward Model (MRM). The numerical results presented by applying models in the SHARPE software package. Furthermore, inter-dependencies between sub-models resolved by the fixed-point iteration technique. Moreover, the analytical results verified by the Discrete-Event Simulation (DES), which conducted in MATLAB. The result could help the MEC providers to assess their platform performance.

Estimating the natural progression of non-invasive ductal carcinoma in situ breast cancer lesions using screening data.

In addition to invasive breast cancer, mammography screening often detects preinvasive ductal carcinoma in situ (DCIS) lesions. The natural progression of DCIS is largely unknown, leading to uncertainty regarding treatment. The natural history of invasive breast cancer has been studied using screening data. DCIS modeling is more complicated because lesions might progress to clinical DCIS, preclinical invasive cancer, or may also regress to a state undetectable by screening. We have here developed a Markov model for DCIS progression, building on the established invasive breast cancer model. We present formulas for the probability of DCIS detection by time since last screening under a Markov model of DCIS progression. Progression rates were estimated by maximum likelihood estimation using BreastScreen Norway data from 1995-2002 for 336,533 women (including 399 DCIS cases) aged 50-69. As DCIS incidence varies by age, county, and mammography modality (digital vs. analog film), a Poisson regression approach was used to align the input data. Estimated mean sojourn time in preclinical, screening-detectable DCIS phase was 3.1 years (95% confidence interval: 1.3, 7.6) with a screening sensitivity of 60% (95% confidence interval: 32%, 93%). No DCIS was estimated to be non-progressive. Most preclinical DCIS lesions progress or regress with a moderate sojourn time in the screening-detectable phase. While DCIS mean sojourn time could be deduced from DCIS data, any estimate of preclinical DCIS progressing to invasive breast cancer must include data on invasive cancers to avoid strong, probably unrealistic, assumptions.

Stochastic Geometry Analysis of Sojourn Time in Multi-Tier Cellular Networks

Impact of mobility will be increasingly important in future generation wireless services and the related challenges will need to be addressed. Sojourn time, the time duration that a mobile user stays within a cell, is a mobility-aware parameter that can significantly impact the performance of mobile users and it can also be exploited to improve resource allocation and mobility management methods in the network. In this paper, we derive the distribution and mean of the sojourn time in multi-tier cellular networks, where spatial distribution of base stations (BSs) in each tier follows an independent homogeneous Poisson point process (PPP). To obtain the sojourn time distribution in multi-tier cellular networks with maximum biased averaged received power association, as the first step, we derive the area of contact, based on which we then derive the linear contact distribution function and chord length distribution of each tier. We also study the relation between mean sojourn time and other mobility-related performance metrics. We show that the mean sojourn time is inversely proportional to the handoff rate, and the complementary cumulative distribution function (CCDF) of sojourn time is bounded from above by the complement of the handoff probability. Moreover, we study the impact of user velocity and network parameters on the sojourn time.

Multistate Models for Examining the Progression of Intermittently Measured Patient-Reported Symptoms Among Patients With Cancer: The Importance of Accounting for Interval Censoring

ContextPatients with cancer in Ontario, Canada, receive symptom monitoring in a standardized fashion using the Edmonton Symptom Assessment System (ESAS). These measurements can be used to understand symptom progression during the cancer trajectory. ObjectivesThis study demonstrates the implementation of multistate models for examining symptom progression, while appropriately accounting for intermittent observation. We also compare the estimates when the panel nature of the data is ignored. MethodsThis was a population-based retrospective cohort study using linked administrative health-care databases. The cohort consisted of patients who were newly diagnosed with a primary cancer and had at least one ESAS assessment completed between 2007 and 2015 in Ontario, Canada. A 5-state model was developed to examine the progression of symptom severity, where estimation was conducted with and without accommodating for the panel nature of the symptom data. ResultsThe study cohort consisted of 212,615 patients diagnosed with cancer, collectively having 1,006,360 ESAS assessments within the first year after diagnosis. The median (interquartile range) of the number of ESAS assessments per patient was 3 (1–6), and the average gap time between consecutive assessments was approximately three months. The estimated mean sojourn time in each state was consistently and significantly greater when ignoring interval censoring than when accounting for it. This held true for all states and symptoms. ConclusionOur work demonstrates the use of multistate models and the importance of accommodating for intermittent observation when examining symptom progression using ESAS among patients with cancer. This work serves as a methodological guide for applied researchers interested in modeling disease progression under the presence of intermittent observation.

Precision Science on Incidence and Progression of Early-Detected Small Breast Invasive Cancers by Mammographic Features.

The aim was to evaluate how the inter-screening interval affected the performance of screening by mammographic appearances. This was a Swedish retrospective screening cohort study with information on screening history and mammography features in two periods (1977–1985 and 1996–2010). The pre-clinical incidence and the mean sojourn time (MST) for small breast cancer allowing for sensitivity by mammographic appearances were estimated. The percentage of interval cancer against background incidence (I/E ratio) was used to assess the performance of mammography screening by different inter-screening intervals. The sensitivity-adjusted MSTs (in years) were heterogeneous with mammographic features, being longer for powdery and crushed stone-like calcifications (4.26, (95% CI, 3.50–5.26)) and stellate masses (3.76, (95% CI, 3.15–4.53)) but shorter for circular masses (2.65, (95% CI, 2.06–3.55)) in 1996–2010. The similar trends, albeit longer MSTs, were also noted in 1977–1985. The I/E ratios for the stellate type were 23% and 32% for biennial and triennial screening, respectively. The corresponding figures were 32% and 43% for the circular type and 21% and 29% for powdery and crushed stone-like calcifications, respectively. Mammography-featured progressions of small invasive breast cancer provides a new insight into personalized quality assurance, surveillance, treatment and therapy of early-detected breast cancer.

Modeling the natural history of ductal carcinoma in situ based on population data

BackgroundThe incidence of ductal carcinoma in situ (DCIS) has increased substantially since the introduction of mammography screening. Nevertheless, little is known about the natural history of preclinical DCIS in the absence of biopsy or complete excision.MethodsTwo well-established population models evaluated six possible DCIS natural history submodels. The submodels assumed 30%, 50%, or 80% of breast lesions progress from undetectable DCIS to preclinical screen-detectable DCIS; each model additionally allowed or prohibited DCIS regression. Preclinical screen-detectable DCIS could also progress to clinical DCIS or invasive breast cancer (IBC). Applying US population screening dissemination patterns, the models projected age-specific DCIS and IBC incidence that were compared to Surveillance, Epidemiology, and End Results data. Models estimated mean sojourn time (MST) in the preclinical screen-detectable DCIS state, overdiagnosis, and the risk of progression from preclinical screen-detectable DCIS.ResultsWithout biopsy and surgical excision, the majority of DCIS (64–100%) in the preclinical screen-detectable state progressed to IBC in submodels assuming no DCIS regression (36–100% in submodels allowing for DCIS regression). DCIS overdiagnosis differed substantially between models and submodels, 3.1–65.8%. IBC overdiagnosis ranged 1.3–2.4%. Submodels assuming DCIS regression resulted in a higher DCIS overdiagnosis than submodels without DCIS regression. MST for progressive DCIS varied between 0.2 and 2.5 years.ConclusionsOur findings suggest that the majority of screen-detectable but unbiopsied preclinical DCIS lesions progress to IBC and that the MST is relatively short. Nevertheless, due to the heterogeneity of DCIS, more research is needed to understand the progression of DCIS by grades and molecular subtypes.

The disasters queue with working breakdowns and impatient customers

We consider theM/M/1 queue with disasters and impatient customers. Disasters only occur when the main server being busy, it not only removes out all present customers from the system, but also breaks the main server down. When the main server is down, it is sent for repair. The substitute server serves the customers at a slow rate(working breakdown service) until the main server is repaired. The customers become impatient due to the working breakdown. The system size distribution is derived. We also obtain the mean queue length of the model and mean sojourn time of a tagged customer. Finally, some performance measures and numerical examples are presented.

Asymptotic Behavior of the Mean Sojourn Time for a Random Walk to be in a Domain of Large Deviations

We study the asymptotic behavior of the mean of sojourn time for a homogeneous random walk defined on $$ [0,n]$$ to be above a receding curvilinear boundary in a domain of large deviations under Cramer’s condition on the jump distribution.

Bartolomeu: An SDN rebalancing system across multiple interdomain paths

Bartolomeu is a solution to enable stub networks to perform adaptive egress traffic load balancing across multiple interdomain routes by spreading the traffic across available paths according to a passive measurement of their performance. It defines a BGP-SDN architecture that increases the number of BGP routes that can be used by stub networks. Bartolomeu measures the available capacity of each path to any destination prefix, and allocates to each path a number of large flows that is proportional to its capacity. This strategy reduces the mean sojourn time, i.e., mean time to flow completion, compared to state-of-the-art traffic balancing techniques as ECMP. We develop a mathematical model to compute this time and compare with ECMP and single path (fast path) selection. An analysis of the traffic traces of two content providers was performed to ensure that our solution is deployable. An experiment with traffic exchange over the Internet is used to show that Bartolomeu can provide gains with real interfering traffic. A discrete-event simulator fed with the traces captured is used to asses Bartolomeu’s gains with prefixes with different number of flows, and flows with different sizes and arrival time. We observe in this experiment that Bartolomeu can reduce the sojourn time, compared to ECMP, by half when path rates differ in a factor of 3, or to a sixth when path rates differ in a factor of 10. We compute the maximum number of per-flow entries and the maximum entry change request rate to show that the resources required fit in with the specifications of the current generation of SDN switches.

Hydrophobic Hydration Processes: Intensity Entropy and Null Thermal Free Energy and Density Entropy and Motive Free Energy.

The processes at the molecule level, which are the source of the ergodic properties of thermodynamic systems, are analyzed with special reference to entropy. The entropy change produced by increasing the temperature T depends on the increase of velocity of the particles with a decrease of the squared mean sojourn time (τm2) and gradual loss of instant energy intensity. The diminution, which is due to dilution, of the number of terms in the summation of cumulative sojourn time (τi2)Σ produces loss of energy density, thus generating a gradual increase of density entropy, dSDens. The ergodic property of thermodynamic systems consists of the equivalence of density entropy (dependent on dilution) with intensity entropy (dependent on temperature). This equivalence has been experimentally verified in every hydrophobic hydration process as thermal equivalent dilution. An ergodic dual-structure partition function {DS-PF} represents the state probability of every hydrophobic hydration process, corresponding to the biphasic composition of these systems. The dual-structure partition function {DS-PF} (Kmot·ζth) is the product of a motive partition function {M-PF} (Kmot) multiplied by a thermal partition function {T-PF} (ζth = 1). {M-PF} gives rise to changes of density entropy, whereas {T-PF} gives rise to changes of intensity entropy. {M-PF} is referred to a reacting mole ensemble (reacting solute) composed of few elements (moles), ruled by binomial distribution, whereas {T-PF} is referred to a nonreacting molecule ensemble (NoremE) (nonreacting solvent), which is composed of a very large population of elements (molecules), ruled by Boltzmann statistics. Statistical thermodynamic methods cannot be applied to {M-PF} that can be calculated by numerical methods from the experimental titration data. By development of the dual-structure partition function {DS-PF}, parabolic convoluted binding functions are obtained. The tangents to the binding functions represent the dual enthalpy, −ΔHdual = (−ΔHmot – ΔHth), and the dual entropy, ΔSdual = (ΔSmot + ΔSth). The connections between canonical and grand-canonical partition functions of statistical thermodynamics with thermal and motive partition functions of chemical thermodynamics, respectively, are discussed. Special attention has been devoted to the equality ΔHth/T + ΔSth = 0, typical of NoremEs, as an entropy–enthalpy compensation with ΔGth/T = 0. The thermodynamic potential change Δμ, as proposed by potential distribution theorem (PDT) for iceberg formation from {T-PF} of the solvent, is nonexistent because the excess solvent is at a constant potential (Δμsolv = 0). The information level offered by the ergodic algorithmic model (EAM) is more complete and correct than that offered by the potential distribution theorem (PDT): the stoichiometry of the water reaction in hydrophobic hydration processes is determined by the EAM as the function of the number ±ξw. Quasi-chemical approximation, renamed the chemical molecule/mole scaling function (Che. m/M. sF), is a fundamental breakthrough in the application of statistical thermodynamics to chemical reactions. Boltzmann statistical molecule distribution of the thermal partition function {T-PF} is scaled with binomial mole distribution of the motive partition function {M-PF}. For computer-assisted drug design, the alternative calculation procedure of Talhout, based on the previous experimental determination of binding functions, is recommended. The ergodic algorithmic model (EAM), applied to the experimental convoluted binding functions, can recover the distinct terms of intensity entropy (ΔHmot/T) and density entropy (ΔSmot), together with other essential information elements, lost by computer simulations.

Quasi-birth and death processes of two-server queues with stalling

This paper investigates an optimal K-policy for a two-server Markovian queueing system $$M/(M_1,M_2)/2/(B_1,B_2),$$ with one fast server $$S_1$$ and one slow server $$S_2$$ , using the matrix analytic method. Two buffers $$B_1$$ and $$B_2$$ are organized to form waiting lines of customers in which, buffer $$B_1$$ is of finite size $$K(< \infty )$$ and buffer $$B_2$$ is of infinite capacity. Buffer $$B_1$$ stalls customers who arrive when the system size (queue + service) is less than $$(K+1)$$ and dispatches a customer to the fast server $$S_1$$ only after $$S_1$$ completes its previous service. This K-policy is of threshold type which deals with controlling of informed customers and hence the customers have better choice of choosing the fast server routing through the buffer $$B_1$$ . The $$(K+2)$$ -nd customer who arrives when the number of customers present in the system is exactly $$(K+1)$$ has the Hobson’s choice of getting service from the slow server $$S_2.$$ Buffer $$B_2$$ accommodates other customers who arrive when the number of customers present in the system is $$(K+2)$$ or more and feeds them one after another to either buffer $$B_1$$ or the sever $$S_2$$ whichever event can first accept the customer at the head-of-the-line in $$B_2$$ . Queue length processes of interest are (1) $$q_1=\lim \limits _{t\rightarrow \infty }X_1(t)$$ and (2) $$q_2=\lim \limits _{t\rightarrow \infty }X_2(t)$$ , where $$X_1$$ (t) represents the number of customers who are in the buffers $$B_1$$ and $$B_2$$ and also in the service with server $$S_1$$ at time ‘t’ and $$X_2$$ (t) represents the number of customers available with server $$S_2$$ only. The bi-variate random sequence $$\mathbf{X}(t)=(X_1(t),X_2(t))$$ of the system size (queue $$+$$ service) forms a quasi-birth and death process (QBD). Steady state characteristics, and some of the performance measures such as the expected queue length, the probability that each server is busy etc are obtained. Numerical illustrations are provided based on the average cost function to explore the methodology of finding the best K-policy which minimizes the mean sojourn time of customers.

The M/G/1 queue system with random vacation

ABSTRACTThe vacation queue models have been studied by many researchers. They usually assume that the server will take a vacation immediately at the end of a busy period. But in some cases, it is random that the server takes a vacation when the system is empty. Therefore, we study an M/G/1 queue model with random vacation policy in this paper. We derive the indexes such as the mean queue length, mean sojourn time of customers and the busy period of the server. Finally, numerical examples are provided for illustration.

Transition Patterns of Weight Status and Their Associated Factors among Elementary School Children: A Longitudinal Cohort Study Using Multistate Markov Model.

Background: Childhood obesity is a recognized public health concern worldwide. It is essential to study the natural progression of obesity in the interest of prevention. This study aimed to describe the dynamic changes in weight status among elementary school children and identify the significant factors influencing the progression or regression of weight status. Methods: This study involved 928 elementary school children who were followed up annually during their elementary school years. Heights, weights, and vital capacity (VC) were measured each school year. A multistate Markov model containing three weight states was fit to longitudinal weight status data. Results: Children with healthy weight and obesity tended to stay in their preceding weight state. Children with overweight, in contrast, were more likely to move to the other two states. The mean sojourn time in obesity and in overweight states was 5.15 years (95% confidence interval [CI]: 4.22-6.3) and 2 years (95% CI: 1.76-2.28), respectively. Children in lower grades, those with a lower VC index, those with a higher initial BMI, those with a higher annual weight increment, and boys were at increased risk of progression to overweight or obesity, with a decreased probability of regression. Conclusions: Children with obesity were more resistant to recovery than those with overweight. Prevention and intervention measures should be adopted early when abnormal weight onset occurs. The multistate Markov model was an advanced tool to analyze dynamic changes in status and identify significant factors for progression and regression and helped to develop prevention and intervention targeting strategies.

Last in line

Queueing is a common praxis in banks, hospitals and transportation, just to name a few. One common performance metric is the mean sojourn time. However, humans experience waiting time in a more complex manner — they dislike being the last in line. We study queueing systems subject to such a cost structure. For the single M/G/1 queue, we derive the corresponding mean costs, value functions and admission costs, which are then applied to route customers to parallel servers.

The Asymptotic Behavior of the Mean Sojourn Time for a Random Walk Above a Receding Curvilinear Boundary

We study the asymptotic behavior of the mean sojourn time for a random walk above a receding curved boundary in the case where the jump distribution satisfies the Cramer condition.

Test sensitivity of mammography and mean sojourn time over 40 years of breast cancer screening in Nijmegen (The Netherlands).

ObjectivesWe investigated whether changes in mammographic technique and screening policy have improved mammographic sensitivity, and elongated the mean sojourn time, since the introduction of biennial breast cancer screening in Nijmegen, the Netherlands, in 1975.MethodsMaximum likelihood estimation, non-linear regression, and Markov Chain Monte Carlo simulation were used to estimate test sensitivity, mean sojourn time, and underlying breast cancer incidence in four time periods, covering 40 years of breast cancer screening in Nijmegen (1975–2012).ResultsMaximum likelihood estimation generated an estimated test sensitivity of approximately 90% and a mean sojourn time around three years, while the estimates based on non-linear regression and Markov Chain Monte Carlo simulation were 80% and four years, respectively. All three methods estimated a rise in the underlying breast cancer incidence over time, with approximately one case more per 1000 women per year in the final period compared with the first period.ConclusionsThe three methods showed a slightly higher mammographic sensitivity and a longer mean sojourn time in the last period, after the introduction of digital mammography. Estimates were more realistic for the more sophisticated methods, non-linear regression and Markov Chain Monte Carlo simulation, while the simple closed form approximation of maximum likelihood estimation led to rather high estimates for sensitivity in the early periods.

Multistate Markov model for predicting the natural disease progression of type 2 diabetes based on hemoglobin A1c

Introduction: Type 2 diabetes is a common non-communicable disease, especially in developing countries like India, posing a huge economic burden on the family and nation as a whole. It is a chronic metabolic disorder in which prevalence has been increasing steadily all over the world. In studies of many chronic medical conditions, the health status of a patient may be characterized using a finite number of disease states. The multi-state Markov model is a useful way to describe states of a disease over time. In this research article, we have illustrated the usefulness of multistate Markov models in the analysis of follow-up of diabetes. The valuable information provided by the hemoglobin A1c (HbA1c) test has rendered it as a reliable biomarker for the diagnosis and prognosis of diabetes. Objectives: The main purpose of this study is to assess the importance and significance of HbA1c as a useful disease marker for type 2 diabetes by using a three-state Markov model. Patients and Methods: A total of 246 type 2 diabetic patients were included in this study. These patients are classified in different states on the basis of their available baseline value of HbA1c. HbA1c repeated after every 1 year for consecutive four years. Based on ranges of HbA1c, three transient states (4 ≤ HbA1C ≤ 5.6, 5.7 ≤ HbA1C ≤ 6.4 and HbA1C ≥ 6.5%) have been defined. Additionally, transition intensities, transition probabilities, mean sojourn time in each state and also expected state specific survival time have been assessed. All the statistical analysis has been performed using the msm package in R software. Results: The mean age of patients at diagnosis was 26.12 years (SD=7.60), ranging from 10 to 49 years. The estimates of transition intensities reveal that a patient in state 1 is 16.4 (0.82/0.05) times more likely to move to state 2 than to move to diabetic state. Similarly, a patient in pre-diabetic is 7.5 (2.34/0.31) times more likely to move to diabetic state as compared to normal state. Additionally, once a patient is in a diabetic state there is 79% chances of remaining in a diabetic state as compared to 4% and 17% of moving to normal or pre-diabetic state, this implies that a patient who once in the diabetic state is difficult to move to a normal or pre-diabetic state. Conclusion: The estimated total length of time spent in each state is forecasted to be four months in normal state, five months in pre-diabetic state and 39 months in diabetic state. Hence, it has been concluded that, once the patient enters the diabetic state (HbA1c>6.4), the chances of getting back to normal or pre-diabetic state are very small.