# Questions tagged [interarrival-time]

The time between the arrival of one point & the next, as in a point process or queue. More abstractly, 'time' can be a distance between 2 points in a space.

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### Strong Renewal Assumption?

Just started a Stochastic Processes course and I am a bit confused over the Strong Renewal Assumption we make for Renewal (and Poisson?) processes. The assumption in my text goes as follows: "At ...
1 vote
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### How to fit exponential and Poisson distributions to table of event times in R?

I have a vector of event dates where multiple events can occur on the same date. I have sorted these dates in chronological order and have also generated a table of times between events. Here is an ...
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### How to simulate events in certain time?

I am trying to make simulation, and I have some events that happened durring a day. Some times it s for example few events durring half hour for one day, and sometimes we have event, and then few ...
1 vote
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### Sanity check for categorising arrivals in groups

I have a question regarding the generation of arrival times. Assuming that I have an arrival rate of e.g. $400$ arrivals per hour. I would like to generate specific arrival times for e.g. $300$ ...
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### Interarrival times using timeseries data

I am attempting to model user requests coming into a computer system. Is there a good way to generate interarrival times for requests using timeseries data? Specifically, my data in the form of ...
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### Why does the superposition of two processes with geometrically distributed interarrival times not result in a process with similar distribution?

Assume we have two random variables $G_1$ and $G_2$ that both follow a geometric distribution of type B (so RVs can assume 0) with success probability $p = p_1 = p_2$ that describe the interarrival ...
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### Non-parametric (smoothed) estimate of current rate

I am looking at time-series event data and need to visualize how the arrival rate $\lambda$ changes over time. I do not want to assume any underlying distribution that the data comes from (it is ...
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### Are arrivals uniformly distributed over a time interval when inter-arrival times are poisson distributed

Assume a fixed time horizon $T$ and suppose that inter-arrivals times at a queue (during the time horizon $T$) are poisson distributed with arrival rate $\lambda$. Will the arrivals be uniformly ...
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1 vote
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### negative binomial inter-arrival

I am trying to find out what is the distribution governing the inter-arrival of events for a counting process that follows a negative binomial? I know for a Poisson counting process the inter-arrival ...
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### Number of events in a segment if waiting times are drawn from a mixture of two exponential distributions

What is the probability for $n$ events to occur over a period of time $t$, if the duration of each event is a random variable distributed as a mixture of two exponential distributions, one with the ...
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### Can the interarrival times of a continuous time markov chain be distributed with 2 parameter (scale,location) exponential distributions?

I'm trying to model data with a time-homogenous CTMC with a number of states with corresponding constant transition rates $\lambda_{i}$ when I notice that much of the transition times from one state ...
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### Conditional distribution of arrival times in Poisson process

Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$. What is the conditional ...
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### How to properly truncate exponential distribution to represent random memory-less arrivals?

Typically, Poisson and exponential distributions are used to represent random memory-less arrival processes. It has come to my attention, however, that a more realistic distribution is a truncated (or ...
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### For binomial success / fail data, how to calculate if the time since last success is out of the normal range

Struggling to figure this out. I have data sets that are essentially ObjectId, Date, Pass/Fail (1/0) The data is normally daily. What I want to figure out is IF the number of days since the last ...
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### two independent Poisson Arrivals

I have two types of customers (type 1 and type 2) enter a shop. Their arrival processes are independent and follow Poisson process with the arrival rates of $\lambda_1$ and $\lambda_2.$ Consider two ...
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### Example of dataset where the data collected at time-points $g(t_1), g(t_2), \ldots$

What would be some practical scenarios where we collect data at time-points $g(t_1), \ldots, g(t_n)$, where $g$ is an increasing function? For example, $g(t) = \exp(t)$ or $\ln t$. To be more clear, ...
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### simulation in R: reading arrival times and service times from a set and have priority queue

I want to simulate an outpatient clinic that does not have walk-in. based on a scheduling model, I have already calculated arrival times and examination times and whether the patient will be a show or ...
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### How to model arrival times in discrete event simulation where arrivals vary with time of day and day of week

I'm looking for suggestions on how to model inter-arrival times in a discrete event simulation where the arrival rate is highly dependent on the day of week and the hour of day. For example: A ...
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### Simulate an arrival time - Cox Process?

If I simulate arrival times using a Poisson process where the input to the Poisson process is also a Poisson process. Input arrival rate = L , and then U1 and U2 are random draws from a uniform ...
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### Poisson process and queuing system

In a queuing system, the inter-arrival times are known to be exponentially distributed. My textbook states: "It can be shown that, if the underlying distribution of inter-arrival times { T1, T2, ..., ...
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### Finite state machine with gamma distributed waiting times

I have a state machine with positive and negative inputs. The time between positive inputs follows a gamma distribution ($X_+ \sim \Gamma(k_+, \theta_+)$), and the time between negative inputs follows ...
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### Poisson Distributions: Have Daily Rate and "peak rate", how to figure out hourly rate?

I know that the daily average number of customers per 11 hours is 610 and during the highest peak of the day (from 12 to 1) 100 customers are served with the second peak (from 6 to 7) 50 customers ...
1 vote
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### Poisson distribution inside-out: estimate the beginning of the period based on first occurrence time

If a Fisherman catches his first fish of the day at 10am, when is he more likely to have started fishing given that in average he catches 1 fish/hour? In other words: Let's assume a Poisson process ...
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### inter arrivals time

I am trying to simulate a user that generates files over a whole year. I have a dataset with information about date and time of files generated from a user and it follows negative binomial ...
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### What is the difference between the Poisson distribution and the uniform distribution?

I'm modeling a factory with orders arriving independently of each other. I need a series of actual time stamps of arrivals. I've been told that everyone uses the Poisson distribution to model arrivals....
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### Interarrival times of exponential distribution

I would like to find the pdf for the $n-1$ interarrival times $w$ of $n$ iid events from an exponential distribution of time constant $\tau$. Using the fact that the first event follows an ...
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### Using stream of page views on a website to detect anomaly or bot-like behavior

I am looking at clickstream data - sequence of page visits of a user in a website. I am trying to identify anomalies in a session that could be related to bots. To do so I am looking at a user's ...
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### How to benchmark an ACD model?

I have trained an Autoregressive Conditional Duration (ACD) model on some transactions data. Then, I test it against some other transactions data to check how well it predicts the time interval ...
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### Calculate predictability of events over time

I'm trying to create a model / algorithm which learns the predictability of events over time, which takes into account both frequency and rarity. An example of what this could apply to (which is the ...
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### Algorithms for extracting moving average intervals

I have a series of discrete events and a global clock. Whenever an event happens, I record a timestamp and do some processing. Specifically, I am trying to compute "moving average" (not really) ...
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### Inter-arrival time of negative binomial distribution

In order to simulate a process I considered two options: number of items received follows a Poisson distribution λ number of items received follows a Negative Binomial distribution (r,p): ...
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### If the mean is equal to the standard deviation, what is the general likelihood that the underlying distribution is normal vs exponential?

This comes from operations analysis. Often you have measurement of times, such as time between customers arriving at a store (e.g. Starbucks) or time it takes to process an order (e.g. a cappuccino). ...
1 vote
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### Fit inter-arrival time to Poisson Distribution/Exponential

I am trying to fit the Poisson distribution to my data which is the inter-arrival times. The format of the data is: time, count 2,3 5,1 8,2 ,which means I had 3 objects with the inter-arrival ...
1 vote
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### Understanding rexp function in R to draw samples from exponential distribution

I want to get the number of arrivals per 1 second for a 10 second period. Let's assume that lambda is 15 per second, and I want a sample size of 10. I generate random numbers exponentially distributed ...
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### How to take incomplete waiting times into account

I am measuring the time between jumps from a jump process. However my observation window starts randomly somewhere in time. Therfore I do not know when the jump before the first observed one occurs. ...
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### Minimum interval such that $X\%$ of intervals will contain $\ge Y$ events

I have a data set that contains the dates and times of purchases. Say I have one month of data for the purchases. I am looking for the interval of time, say 10 mins / 30 mins / hourly, for which 90% ...
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### Normalizing year data

I have a data set in which there are data from April 2010 to Dec 2010 ( 9 Months) Jan 2011 to Dec 2011 (12 Months ) Jan 2012 to Dec 2012 (12 Months) Jan 2013 to Dec 2013 (12 Months) Jan 2014 to ...
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### Are Poisson distributions only for arrivals?

I have this data set: ...
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### Modeling Arrivals With a Time Limit

I have some data (sample here): ...
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### Treatment of waiting time data with repeated observations for a finite time

A call center is open for 9 hours every day. My data are the times of occurrence of calls, divided in days. Some days are busier than the others and some days no calls at all occur. I have to estimate ...
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### What distributions are possible for an arrival rate?

I'm really struggling to find a good statistical distribution. I've tried Poisson and Gamma so far, but without success (best I've got was a p-value of 0,00005 with a Pearson Chi-Square test). So I ...
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### Survival Analysis/Frailty Model

I am trying to build a probability model for the following situation: I have a very large dataset, whose rows are users and columns are the features. All the users visit a particular store X (once ...
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### Is there a family of processes centred on the Poisson process?

I am looking for a model, characterized continuously by a single parameter, to describe the arrival times of buses with unit expected interarrival time. At one extreme of the parameter (say $\theta=1$)...
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### Predicting on data consisting of many independent short time series?

I have a dataset which contains many (hundreds) of short (3-30 obs) timeseries of different lengths. Each series is currently represented as a number indicating how long it took to the next event. To ...
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### Estimating Poisson process intensity using GLM

Suppose I want to build an explanatory model for events generated by an inhomogeneous Poisson process with unknown intensity $\lambda$. Each entry in my dataset represents the registration of an event:...
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### Interarrival time distribution of uniform arrival process

I am currently trying to model an uniform arrival process within my simulation model. However, I can only model it by means of an interarrival time (I can let the model wait for a certain amount ...
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### Models for learning acquistion process

Imagine such a memory test for a mice. The mice performs an experiment $E$ with two possible issues $0$ (failure) and $1$ (success). If the mice gets $1$ it is "rewarded", if it gets $0$ it is "...
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### Can I estimate the parameter of a Poisson arrival process from a low-incidence observation period?

If I know only that the arrival process is Poisson, and I observe it for a pre-chosen (say, unit) period of time, observing $k$ arrivals, is it meaningful to describe an estimate of its time parameter ...
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