Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Waiting-Time Paradox
57
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Suppose that youre going to school and arrive at a bus stop. How long do you have to wait before the next bus arrives? Surprisingly, it is longer - possibly much longer - than what the bus schedule suggests intuitively. This phenomenon, which is called the waiting-time paradox, has a purely mathematical origin. Different buses arrive with different intervals, leading to this paradox. In this article, we explore the waiting-time paradox, explain why it happens, and discuss some of its implications (beyond the possibility of being late for school).
rate research
Read More
This mathematical recreation extends the analysis of a recent paper, asking when a traveller at a bus stop and not knowing the time of the next bus is best advised to wait or to start walking toward the destination. A detailed analysis and solution is provided for a very general class of probability distributions of bus arrival time, and the solution characterised in terms of a function analogous to the hazard rate in reliability theory. The note also considers the question of intermediate stops. It is found that the optimal strategy is not always the laziest, even when headways are not excessively long. For the common special case where one knows the (uniform) headway but not the exact timetable, it is shown that one should wait if the headway is less than the walking time (less bus travel time), and walk if the headway is more than twice this much. In between it may be better to wait or to walk, depending on ones confidence in being able to catch up to a passing bus.
We consider a model of a population of fixed size N in which each individual gets replaced at rate one and each individual experiences a mutation at rate mu. We calculate the asymptotic distribution of the time that it takes before there is an individual in the population with m mutations. Several different behaviors are possible, depending on how mu changes with N. These results have applications to the problem of determining the waiting time for regulatory sequences to appear and to models of cancer development.
In high frequency financial data not only returns but also waiting times between trades are random variables. In this work, we analyze the spectra of the waiting-time processes for tick-by-tick trades. The numerical problem, strictly related with the real inversion of Laplace transforms, is analyzed by using Tikhonovs regularization method. We also analyze these spectra by a rough method using a comb of Diracs delta functions.
Few statistically compelling correlations are found in pulsar timing data between the size of a rotational glitch and the time to the preceding glitch (backward waiting time) or the succeeding glitch (forward waiting time), except for a strong correlation between sizes and forward waiting times in PSR J0537-6910. This situation is counterintuitive, if glitches are threshold-triggered events, as in standard theories (e.g. starquakes, superfluid vortex avalanches). Here it is shown that the lack of correlation emerges naturally, when a threshold trigger is combined with secular stellar braking slower than a critical, calculable rate. The Pearson and Spearman correlation coefficients are computed and interpreted within the framework of a state-dependent Poisson process. Specific, falsifiable predictions are made regarding what objects currently targeted by long-term timing campaigns should develop strong size-waiting-time correlations, as more data are collected in the future.
Multiserver jobs, which are jobs that occupy multiple servers simultaneously during service, are prevalent in todays computing clusters. But little is known about the delay performance of systems with multiserver jobs. We consider queueing models for multiserver jobs in a scaling regime where the total number of servers in the system becomes large and meanwhile both the system load and the number of servers that a job needs scale with the total number of servers. Prior work has derived upper bounds on the queueing probability in this scaling regime. However, without proper lower bounds, the existing results cannot be used to differentiate between policies. In this paper, we study the delay performance by establishing sharp bounds on the mean waiting time of multiserver jobs, where the waiting time of a job is the time spent in queueing rather than in service. We first consider the commonly used First-Come-First-Serve (FCFS) policy and characterize the exact order of its mean waiting time. We then prove a lower bound on the mean waiting time of all policies, and demonstrate that there is an order gap between this lower bound and the mean waiting time under FCFS. We finally complement the lower bound with an achievability result: we show that under a priority policy that we call P-Priority, the mean waiting time achieves the order of the lower bound. This achievability result implies the tightness of the lower bound, the asymptotic optimality of P-Priority, and the strict suboptimality of FCFS.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
