Average Sojourn Time Research Articles

Matched queues with matching batch pair [formula omitted]

In this paper, we develop the matrix-analytic method to discuss an interesting but challenging bilateral stochastic matching problem: A matched queue with matching batch pair (m,n) and two types of impatient customers, where the two types of customers arrive according to two independent Poisson processes. Once m A-customers and n B-customers are matched as a group, the m+n customers immediately leave the system. We show that this matched queue can be expressed as a novel bidirectional level-dependent quasi-birth-and-death (QBD) process whose analysis has its own interests, and specifically, computing the maximal non-positive inverse matrices of bidirectional infinite sizes by using the RG-factorizations. Based on this, we can provide an effective matrix-analytic method to deal with this matched queue, including the system stability, the average stationary queue lengths, the average sojourn times, and the departure process. We believe that the methodology and results developed in this paper can be applicable to studying more general matched queueing systems, which are widely encountered in many practical areas.

On the Optimal Input Rate in Queues with Batch Service

In recent years, queuing systems with batch service are emerging as powerful and flexible mathematical models in different frameworks. In this paper, we consider a single server queuing system with Poissonian arrivals, infinite buffers, and a constant batch size b. This paper addresses a little-studied optimization problem, namely the existence of an optimal arrival rate that minimizes the average sojourn time. Unlike the classical M/M/1 queue, for any batch size b, the problem admits a non-trivial solution that can be found by solving a polynomial equation of degree b+1. Since, in general, only numerical solutions are available, a simple first-order approximation is also derived and the corresponding deviations (in terms of input rate and sojourn time) are calculated. In more detail, it is shown that the approximation improves as the batch size increases and, in any case, the relative error for the average sojourn time is less than 0.34%. Finally, the paper provides new theoretical results about the asymptotic service rate in the equivalent birth–death process, highlighting how it depends on all queue parameters.

Synchronization of semi-Markovian switching complex networks with partial unknown transition rates via event-triggered control

This article investigates the synchronization of semi-Markovian switching complex networks with partial unknown transition rates. Different from Markovian switching system, semi-Markov process is characterized by a fixed transition probability matrix and a probability density function of the sojoturn time, and the sojourn time obeys an expanded exponential distribution, which relaxes the restriction that the sojoturn time obeys memoryless exponential distribution. To realize synchronization, event-triggered control is applied which effectively reduced the control cost. It is demonstrated that the semi-Markovian switching complex networks can realize synchronization under appropriate control gain, event-triggered parameter and average sojourn time. Besides, to cope with the uncertainties that may occur during the switching process, the synchronization of the semi-Markovian switching complex networks with parameter uncertainties and external disturbances is also discussed. In the end, two simulation examples are provided to verify the feasibility of our results.

Coordinating supply and demand in ride-sourcing markets with pre-assigned pooling service and traffic congestion externality

This paper proposes an equilibrium model to describe a ride-sourcing market with both ride-pooling (RP) and non-pooling (NP) services considering traffic congestion externality. In particular, we develop a deductive model to approximate the pool-matching probability and passengers’ expected waiting time under a pre-assigned pool-matching mechanism with meeting points, which is being used in products like Uber Express Pool. Using the model, we derive the equilibrium properties and examine the optimal operating strategies for achieving two market scenarios, including the monopoly optimum and social optimum. In contrast to the traditional wisdom, we find that the market could exist in one or multiple equilibria, depending on the trip fare difference between RP and NP services. Additionally, we show that average sojourn time first decreases in the vehicle fleet size (when traffic congestion is mild) and then increases in the vehicle fleet size (when traffic congestion is heavy), exhibiting a “smiling curve”. It is found that the monopoly optimum solution is always located in the downward side of the “smiling curve”, which implies the ride-sourcing platform will self-regulate its fleet size to avoid heavy traffic congestion. We also discuss the impacts of potential labor supply and background traffic on the platform’s optimal operating strategy and its optimal profit. This study gains interesting managerial insights that help understand how ride-sourcing platforms determine optimal operating strategies for both RP and NP services by considering traffic congestion externality.

Optimization of Open Queuing Networks with Batch Services

In this paper, open queuing networks with Poisson arrivals and single-server infinite buffer queues are considered. Unlike traditional queuing models, customers are served (with exponential service time) in batches, so that the nodes are non-work-conserving. The main contribution of this work is the design of an efficient algorithm to find the batch sizes which minimize the average response time of the network. As preliminary steps at the basis of the proposed algorithm, an analytical expression of the average sojourn time in each node is derived, and it is shown that this function, depending on the batch size, has a single minimum. The goodness of the proposed algorithm and analytical formula were verified through a discrete-event simulation for an open network with a non-tree structure.

Double-ended queues with non-Poisson inputs and their effective algorithms

It is interesting and challenging to study double-ended queues with First-Come-First-Match discipline under customers’ impatient behavior and non-Poisson inputs. The system stability can be guaranteed by the customers’ impatient behavior, while the existence of impatient customers makes analysis of such double-ended queues more difficult or even impossible to find an explicitly analytic solution, thus it becomes more and more important to develop effective numerical methods in a variety of practical matching problems. This paper studies a block-structured double-ended queue, whose block structure comes from two independent Markovian arrival processes (MAPs), which are non-Poisson inputs. We show that such a queue can be expressed as a new bilateral quasi birth-and-death (QBD) process which has its own interest. Based on this, we provide a detailed analysis for both the bilateral QBD process and the double-ended queue, including the system stability, the queue size distributions, the average stationary queue lengths, and the sojourn time of any arriving customers. Furthermore, we develop three effective algorithms for computing the performance measures (i.e., the probabilities of stationary queue lengths, the average stationary queue lengths, and the average sojourn times) of the double-ended queue with non-Poisson inputs. Finally, we use some numerical examples in tabular and graphical to illustrate how the performance measures are influenced by some key system parameters. We believe that the methodology and results described in this paper can be applicable to deal with more general double-ended queues in practice, and develop some effective algorithms for the purpose of many actual uses.

The Effects of Prognostic Factors on Metastasis and Survival of Patients with Breast Cancer Using a Multi-State Model

Background: The multi-state models help more closely study of the factors affecting the survival of patients with breast cancer. Method: We conducted the present retrospective cohort study on 2030 Iranian patients with breast cancer in 2020. The patients’ follow-up period ranged from 1 month to 15 years. Accordingly, the initial treatment, metastasis, and death were considered as the first, second, and absorbing states, respectively. The multi-state model was utilized for modeling and analyzing the data at a 95% significance level using the MSM package in R software. Results: The mean age (±SD) of the patients included at diagnosis time was 55.3 (±12.07) years old. The first one year and 5 years adjusted transition probabilities for transitions from the treatment to metastasis estimated as 0.85 (0.15 – 0.89) and 0.45 (0.21 – 0.61), and for metastasis to death transitions, they were estimated as 0.15 (0.1 – 0.21) and 0.55 (0.41 - 0.69), respectively. Moreover, the average sojourn times were estimated as 0.27 and 74.85 months for the treatment and metastasis states, respectively. Conclusion: The obtained results revealed that over time, the transition probabilities of patients from surgery to metastasis state decreased whereas the transition probabilities from metastasis to death state increased using the multi-state model.

Pooled vs. Dedicated Queues when Customers Are Delay-Sensitive

It is generally accepted that operating with a combined (i.e., pooled) queue rather than separate (i.e., dedicated) queues is beneficial because pooling queues reduces long-run average sojourn time. In fact, this is a well-established result in the literature when jobs cannot make decisions and servers and jobs are identical. An important corollary of this finding is that pooling queues improves social welfare in the aforementioned setting. We consider an observable multiserver queueing system that can be operated with either dedicated queues or a pooled one. Customers are delay-sensitive, and they decide to join or balk based on queue length information upon arrival; they are not subject to an external admission control. In this setting, we prove that, contrary to the common understanding, pooling queues can increase the long-run average sojourn time so much that the pooled system results in strictly smaller social welfare (and strictly smaller consumer surplus) than the dedicated system under certain conditions. Specifically, pooling queues hurts performance when the arrival-rate-to-service-rate ratio is large (e.g., greater than one) and the normalized service benefit is also large. We prove that the performance loss due to pooling queues can be significant. Our numerical studies demonstrate that pooling queues can decrease the social welfare (and consumer surplus) by more than 95%. The benefit of pooling is commonly believed to increase with system size. In contrast, we show that when delay-sensitive customers make rational joining decisions, the magnitude of the performance loss due to pooling can strictly increase with the system size. This paper was accepted by Terry Taylor, operations management.

Hybrid DES-PSO framework for the design of commuters’ circulation space at multimodal transport interchange

The commuters’ circulation space in the multimodal transport interchanges comprises of a network of pathways and stairways and meant for the efficient transit of commuters from one mode to another transport mode. Transit Cooperative Research Program (TCRP) - Report 165 guidelines for the design of commuters’ circulation space in multimodal transport interchanges has several shortcomings. To aid the designers of multimodal transport interchanges, a Discrete-Event Simulation (DES) framework using Phase-Type (PH) distribution is developed for the analysis of commuters’ circulation space, based on commuters’ flow–density​ relationship. The performance measures such as average area occupied per commuters, blocking probability, average sojourn time and throughput are computed. The DES framework is then coupled with the Particle Swarm Optimization (PSO) to create a hybrid DES-PSO framework for the design of commuters’ circulation space under a required Level of Service (LOS). Experimental results show that the optimized widths are highly affected by both commuters’ arrival rates and randomness in the arrival flow. The commuters’ flow–density relationship considers both congested and uncongested conditions and therefore gives higher optimized width. The upstream stairways resulted in larger widths than elevated passageways due to slower movement of commuters. The sensitivity analysis under several factors reveals that commuters’ arrival rate and widths of the circulation space are highly dominant factors while the lengths of circulation space do not show any vital impact on performance measure values.

Joint analysis of duration of ventilation, length of intensive care, and mortality of COVID-19 patients: a multistate approach

BackgroundThe clinical progress of patients hospitalized due to COVID-19 is often associated with severe pneumonia which may require intensive care, invasive ventilation, or extracorporeal membrane oxygenation (ECMO). The length of intensive care and the duration of these supportive therapies are clinically relevant outcomes. From the statistical perspective, these quantities are challenging to estimate due to episodes being time-dependent and potentially multiple, as well as being determined by the competing, terminal events of discharge alive and death.MethodsWe used multistate models to study COVID-19 patients’ time-dependent progress and provide a statistical framework to estimate hazard rates and transition probabilities. These estimates can then be used to quantify average sojourn times of clinically important states such as intensive care and invasive ventilation. We have made two real data sets of COVID-19 patients (n = 24* and n = 53**) and the corresponding statistical code publically available.ResultsThe expected lengths of intensive care unit (ICU) stay at day 28 for the two cohorts were 15.05* and 19.62** days, while expected durations of mechanical ventilation were 7.97* and 9.85** days. Predicted mortality stood at 51%* and 15%**. Patients mechanically ventilated at the start of the example studies had a longer expected duration of ventilation (12.25*, 14.57** days) compared to patients non-ventilated (4.34*, 1.41** days) after 28 days. Furthermore, initially ventilated patients had a higher risk of death (54%* and 20%** vs. 48%* and 6%**) after 4 weeks. These results are further illustrated in stacked probability plots for the two groups from time zero, as well as for the entire cohort which depicts the predicted proportions of the patients in each state over follow-up.ConclusionsThe multistate approach gives important insights into the progress of COVID-19 patients in terms of ventilation duration, length of ICU stay, and mortality. In addition to avoiding frequent pitfalls in survival analysis, the methodology enables active cases to be analyzed by allowing for censoring. The stacked probability plots provide extensive information in a concise manner that can be easily conveyed to decision makers regarding healthcare capacities. Furthermore, clear comparisons can be made among different baseline characteristics.

A change point control chart for monitoring the production lead time based on sum of squared ranks

Assuming that a two-stage serial-parallel processing system is an M/G/m queuing system. Firstly, let sojourn time of the order in one stage is the system state, and the state space equation is used to model this system. Also, the influence of the mean shift of sojourn time on the downstream stage is analyzed. Secondly, the average sojourn time is calculated. And the distribution function of the sojourn time is estimated based on the Laplace-Stieltjes (LS) transformation. Thirdly, a change point control chart is designed based on the sum of squared ranks to monitor back order. Finally, the performance of the change point control chart is evaluated via simulation in terms of average run length, the accuracy of the change point estimation and the out-of-control stage estimation. The results show that the change point control chart can effectively detect the mean shift of the lead time, and it can estimate the change point and its out-of-control stage more accurately.

Blood Pressure States Transition Inference Based on Multi-State Markov Model.

The investigation of risk factors associated with hypertension patients has been extensively studied in the past decades. However, the pattern of natural progressive trajectories to hypertension from nonhypertensive states was rarely explored. In this study, we are interested in discovering the underlying transition patterns between different blood pressure states, namely normal state, elevated state, and hypertensive state among the working population in the United States. A multi-state Markov model was built based on 88,966 clinical records from 34,719 participants we collected during the worksite preventive screening from 2012 to 2018. We first investigated the various risk factors, and we found that body mass index (BMI) is the most critical factor for developing new-onset hypertension. The transition probabilities, survival probabilities, and sojourn time of each state were derived given different levels of BMI, age groups, and gender categories. We found the underweight participants are more likely to remain in the current nonhypertensive states within 3 years, while extremely obese participants have a higher probability of developing hypertension. We discovered the distinct transition patterns among male and female participants. On average, the sojourn time in the normal state for normal-weight participants is 4.33 years for females and 2.18 years for their male counterparts. For the extremely obese participants, the average sojourn time in the normal state is 1.38 years for females and 0.71 years for males. In the end, a web-based graphical user interface (GUI) application was developed for clinicians to visualize the impact of behavioral interventions on delaying the progression of hypertension. Our analysis can provide a unique insight into hypertension research and proactive interventions.

Service facilities with risk‐averse customers: a simulation approach

AbstractIn this study, a one‐dimensional cellular automata model is used to represent a self‐organized queueing system with local interaction between captive and boundedly rational customers who repeatedly choose a facility for service. While previous work has focused on decision rules based on adaptive expectations, the present work expands this analysis by explicitly incorporating customers’ attitude toward risk to study the impact of risk aversion on the collective behavior and the average system sojourn time. The customers’ decision process is modeled using adaptive expectations and incorporating the uncertainty involved in these expectations. Customers update their expectations based on their own experience and that of their neighbors. Simulation analysis is used to compare the aggregated behavior for different degrees of customer risk aversion. Risk‐neutral customers base their decisions only on their expected sojourn time, while risk‐averse customers account for uncertainty by estimating an upper bound of the sojourn times. The results indicate that the more risk‐averse the customers, the longer the transient period, and the more slowly the system converges to an almost stable average sojourn time. Systems where customers have an intermediate level of risk aversion achieve the worst average sojourn times.

Efficient multi-objective algorithm for the lot-streaming hybrid flowshop with variable sub-lots

Abstract Recent years, the multi-objective evolutionary algorithm based on decomposition (MOEA/D) has been researched and applied for numerous optimization problems. In this study, we propose an improved version of MOEA/D with problem-specific heuristics, named PH-MOEAD, to solve the hybrid flowshop scheduling (HFS) lot-streaming problems, where the variable sub-lots constraint is considered to minimize four objectives, i.e., the penalty caused by the average sojourn time, the energy consumption in the last stage, as well as the earliness and the tardiness values. For solving this complex scheduling problem, each solution is coded by a two-vector-based solution representation, i.e., a sub-lot vector and a scheduling vector. Then, a novel mutation heuristic considering the permutations in the sub-lots is proposed, which can improve the exploitation abilities. Next, a problem-specific crossover heuristic is developed, which considered solutions with different sub-lot size, and therefore can make a solution feasible and enhance the exploration abilities of the algorithm as well. Moreover, several problem-specific lemmas are proposed and a right-shift heuristic based on them is subsequently developed, which can further improve the performance of the algorithm. Lastly, a population initialization mechanism is embedded that can assign a fit reference vector for each solution. Through comprehensive computational comparisons and statistical analysis, the highly effective performance of the proposed algorithm is favorably compared against several presented algorithms, both in solution quality and population diversity.

Use of personalised risk-based screening schedules to optimise workload and sojourn time in screening programmes for diabetic retinopathy: A retrospective cohort study.

BackgroundNational guidelines in most countries set screening intervals for diabetic retinopathy (DR) that are insufficiently informed by contemporary incidence rates. This has unspecified implications for interval disease risks (IDs) of referable DR, disparities in ID between groups or individuals, time spent in referable state before screening (sojourn time), and workload. We explored the effect of various screening schedules on these outcomes and developed an open-access interactive policy tool informed by contemporary DR incidence rates.Methods and findingsScottish Diabetic Retinopathy Screening Programme data from 1 January 2007 to 31 December 2016 were linked to diabetes registry data. This yielded 128,606 screening examinations in people with type 1 diabetes (T1D) and 1,384,360 examinations in people with type 2 diabetes (T2D). Among those with T1D, 47% of those without and 44% of those with referable DR were female, mean diabetes duration was 21 and 23 years, respectively, and mean age was 26 and 24 years, respectively. Among those with T2D, 44% of those without and 42% of those with referable DR were female, mean diabetes duration was 9 and 14 years, respectively, and mean age was 58 and 52 years, respectively. Individual probability of developing referable DR was estimated using a generalised linear model and was used to calculate the intervals needed to achieve various IDs across prior grade strata, or at the individual level, and the resultant workload and sojourn time. The current policy in Scotland—screening people with no or mild disease annually and moderate disease every 6 months—yielded large differences in ID by prior grade (13.2%, 3.6%, and 0.6% annually for moderate, mild, and no prior DR strata, respectively, in T1D) and diabetes type (2.4% in T1D and 0.6% in T2D overall). Maintaining these overall risks but equalising risk across prior grade strata would require extremely short intervals in those with moderate DR (1–2 months) and very long intervals in those with no prior DR (35–47 months), with little change in workload or average sojourn time. Changing to intervals of 12, 9, and 3 months in T1D and to 24, 9, and 3 months in T2D for no, mild, and moderate DR strata, respectively, would substantially reduce disparity in ID across strata and between diabetes types whilst reducing workload by 26% and increasing sojourn time by 2.3 months. Including clinical risk factor data gave a small but significant increment in prediction of referable DR beyond grade (increase in C-statistic of 0.013 in T1D and 0.016 in T2D, both p < 0.001). However, using this model to derive personalised intervals did not have substantial workload or sojourn time benefits over stratum-specific intervals. The main limitation is that the results are pertinent only to countries that share broadly similar rates of retinal disease and risk factor distributions to Scotland.ConclusionsChanging current policies could reduce disparities in ID and achieve substantial reductions in workload within the range of IDs likely to be deemed acceptable. Our tool should facilitate more rational policy setting for screening.

Content Placement and User Association for Delay Minimization in Small Cell Networks

In small cell networks, content placement can reduce the content download delay at backhaul links while imposing challenges on user association problems. Specifically, a user may be associated with a small base station (SBS) that has the desired contents but is far away, increasing the delay over radio access links. To this end, we investigate the joint content placement and user association problem to reduce the average download delay over backhaul and radio access links for content delivery. Considering both dynamic user population and natural tandem systems, we model content delivery in each SBS as a tandem queue and thus derive the average sojourn time (the average time contents stay in the tandem queue), i.e., the average download delay. On this basis, we formulate a delay minimization problem, encompassing user characteristics (traffic arrival rates, requested content lengths and content preferences) and SBS constraints (backhaul capacity and storage size). Then, this problem is decomposed into a user association subproblem and a content placement subproblem by exploiting the biconvexity. Finally, a distributed content placement and user association scheme is proposed by exploring the optimal match with the first derivative length method. Extensive simulation results reveal that the delay under the proposed content placement policy may be reduced as user characteristics become heterogeneous, which is opposite to the case under user association policies.

An Energy Efficient Task Scheduling Strategy in a Cloud Computing System and its Performance Evaluation using a Two-Dimensional Continuous Time Markov Chain Model

With ongoing energy shortages and rises in greenhouse emissions worldwide, increasing academic attention is being turned towards ways to improve the efficiency and sustainability of cloud computing. In this paper, we present a performance analysis and a system optimization of a cloud computing system with an energy efficient task scheduling strategy directed towards satisfying the service level agreement of cloud users while at the same time improving the energy efficiency in cloud computing system. In this paper, we propose a novel energy-aware task scheduling strategy based on a sleep-delay timer and a waking-up threshold. To capture the stochastic behavior of tasks with the proposed strategy, we establish a synchronous vacation queueing system combining vacation-delay and N-policy. Taking into account the total number of tasks and the state of the physical machine (PM), we construct a two-dimensional continuous-time Markov chain (CTMC), and produce an infinitesimal generator. Moreover, by using the geometric-matrix solution method, we analyze the queueing model in the steady state, and then, we derive the system performance measures in terms of the average sojourn time and the energy conservation level. Furthermore, we conduct system experiments to investigate the proposed strategy and validate the system model according to performance measures. Statistical results show that there is a compromise between the different performance measures when setting strategy parameters. By combining different performance measures, we develop a cost function for the system optimization. Finally, by dynamically adjusting the crossover probability and the mutation probability, and initializing the individuals with chaotic equations, we present an improved genetic algorithm to jointly optimize the sleep parameter, the sleep-delay parameter and the waking-up threshold.

Resource Allocation for High-Reliability Low-Latency Vehicular Communications With Packet Retransmission

In vehicular communications, safety-critical information exchange has stringent reliability and latency requirements. However, lack of accurate instantaneous channel state information due to high mobility poses a great challenge to meet these requirements and the situation gets more complicated when packet retransmission is considered. Based on only large-scale fading channel information, we perform spectrum and power allocation to maximize the sum ergodic capacity of vehicular-to-infrastructure (V2I) links, while guaranteeing reliability and latency requirements of vehicular-to-vehicular (V2V) links. First, for each possible V2I–V2V spectrum reusing pair, we analyze the queueing system for the V2V link and derive the packet loss probability and average packet sojourn time under fixed power allocation. Then, an optimal power allocation is derived for each possible pair. Afterwards, we optimize the spectrum reusing pattern by addressing a polynomial time solvable bipartite matching problem. Simulation results confirm the accuracy of the proposed queueing analysis. Moreover, the proposed resource allocation strategy can improve the ergodic capacity of the V2I links under the precondition of guaranteeing the V2V links’ requirements on reliability and latency.

Estimands to quantify prolonged hospital stay associated with nosocomial infections

BackgroundLength of stay evaluations are very common to determine the burden of nosocomial infections. However, there exist fundamentally different methods to quantify the prolonged length of stay associated with nosocomial infections. Previous methodological studies emphasized the need to account for the timing of infection in order to differentiate the length of stay before and after the infection.MethodsWe derive four different approaches in a simple multi-state framework, display their mathematical relationships in a multiplicative as well as additive way and apply them to a real cohort study (n=756 German intensive-care unit patients of whom 124 patients acquired a nosocomial infection).ResultsThe first approach ignores the timing of infection and quantifies the difference of eventually infected and eventually uninfected; it is 12.31 days in the real data. The second approach compares the average sojourn time with infection with the average sojourn time of being hypothetically uninfected; it is 2.12 days. The third one compares the average length of stay of a population in a world with nosocomial infections with a population in a hypothetical world without nosocomial infections; it is 0.35 days. Finally, approach four compares the mean residual length of stay between currently infected and uninfected patients on a daily basis; the difference is 1.77 days per infected patient.ConclusionsThe first approach should be avoided because it compares the eventually infected with the eventually uninfected, but has no prospective interpretation. The other approaches differ in their interpretation but are suitable because they explicitly distinguish between the pre- and post-time of the nosocomial infection.

Strategic Behavior of Cognitive Radio Networks With Different Information

The equilibrium joining probabilities for a single primary user (PU) and secondary users (SUs) in the case of no queue length information, and the equilibrium Nash balance thresholds for PU and SUs to join the system in the cases of partial queue length information and full queue length information observed by PU and SUs are investigated in a cognitive radio system. PU stochastically sends PU requests and each SU is assumed to only carry one SU request. Whenever a PU (or an SU) request is completed, a reward is issued to the PU (or the SU). However, a holding cost is charged also for each SU and PU during their average sojourn time in the system respectively. Both of PU requests and SU requests are selfish in this research and their objectives are to maximize their own benefit. Our major conclusion includes a threshold probability value for PU (or SU) to join the system identified for the case of no queue length information, a Nash balance PU (SU) threshold value is determined for the case of partial queue length information; and the multi-dimensional Nash balance threshold is established for the case of full queue length information. These Nash balance threshold integer values depend on not only the Nash Balance PU-threshold value but also the specific number of SU requests in the system at that arriving time. In addition, we discuss the sensibility of input parameters to the obtained equilibrium joining probabilities and Nash balance thresholds. We mathematically verify that SUs’ equilibrium joining probability does not necessarily increase with the transmission rate of PU in no queue length information case and observe numerically that the Nash balance thresholds adopted by an arriving SU are fickle with the change of PU's transmission rate in both of partial and full queue length information cases. Our results and observations provide a guideline and important managerial insights in making decisions with strategic users in the design of CRNs.