Thanks for contributing an answer to Cross Validated! A coin lands heads with chance $p$. The average wait for an interval of length $15$ is of course $7\frac{1}{2}$ and for an interval of length $45$ it is $22\frac{1}{2}$. However, in case of machine maintenance where we have fixed number of machines which requires maintenance, this is also a fixed positive integer. The typical ones are First Come First Served (FCFS), Last Come First Served (LCFS), Service in Random Order (SIRO) etc.. We can find this is several ways. Another way is by conditioning on $X$, the number of tosses till the first head. E gives the number of arrival components. Examples of such probabilistic questions are: Waiting line modeling also makes it possible to simulate longer runs and extreme cases to analyze what-if scenarios for very complicated multi-level waiting line systems. Also make sure that the wait time is less than 30 seconds. Why did the Soviets not shoot down US spy satellites during the Cold War? The best answers are voted up and rise to the top, Not the answer you're looking for? Asking for help, clarification, or responding to other answers. How can I change a sentence based upon input to a command? Here are the expressions for such Markov distribution in arrival and service. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Introduction. Your home for data science. You need to make sure that you are able to accommodate more than 99.999% customers. )=\left(\int_{yx}xdy\right)=15x-x^2/2$$ Can I use a vintage derailleur adapter claw on a modern derailleur. &= \sum_{k=0}^\infty\frac{(\mu t)^k}{k! I think the approach is fine, but your third step doesn't make sense. Your simulator is correct. Let's call it a $p$-coin for short. The number of distinct words in a sentence. So $W$ is exponentially distributed with parameter $\mu-\lambda$. Not everybody: I don't and at least one answer in this thread does not--that's why we're seeing different numerical answers. It uses probabilistic methods to make predictions used in the field of operational research, computer science, telecommunications, traffic engineering etc. Stochastic Queueing Queue Length Comparison Of Stochastic And Deterministic Queueing And BPR. service is last-in-first-out? By Ani Adhikari Tavish Srivastava, co-founder and Chief Strategy Officer of Analytics Vidhya, is an IIT Madras graduate and a passionate data-science professional with 8+ years of diverse experience in markets including the US, India and Singapore, domains including Digital Acquisitions, Customer Servicing and Customer Management, and industry including Retail Banking, Credit Cards and Insurance. Round answer to 4 decimals. That they would start at the same random time seems like an unusual take. This means that there has to be a specific process for arriving clients (or whatever object you are modeling), and a specific process for the servers (usually with the departure of clients out of the system after having been served). the $R$ed train is $\mathbb{E}[R] = 5$ mins, the $B$lue train is $\mathbb{E}[B] = 7.5$ mins, the train that comes the first is $\mathbb{E}[\min(R,B)] =\frac{15}{10}(\mathbb{E}[B]-\mathbb{E}[R]) = \frac{15}{4} = 3.75$ mins. Your expected waiting time can be even longer than 6 minutes. The average response time can be computed as: The average time spent waiting can be computed as follows: To give a practical example, lets apply the analysis on a small stores waiting line. For definiteness suppose the first blue train arrives at time $t=0$. If $W_\Delta(t)$ denotes the waiting time for a passenger arriving at the station at time $t$, then the plot of $W_\Delta(t)$ versus $t$ is piecewise linear, with each line segment decaying to zero with slope $-1$. The logic is impeccable. Sincerely hope you guys can help me. if we wait one day $X=11$. This gives a expected waiting time of $\frac14 \cdot 7.5 + \frac34 \cdot 22.5 = 18.75$. The method is based on representing \(W_H\) in terms of a mixture of random variables. This waiting line system is called an M/M/1 queue if it meets the following criteria: The Poisson distribution is a famous probability distribution that describes the probability of a certain number of events happening in a fixed time frame, given an average event rate. This is a M/M/c/N = 50/ kind of queue system. We can find $E(N)$ by conditioning on the first toss as we did in the previous example. We can expect to wait six minutes or less to see a meteor 39.4 percent of the time. i.e. This is a shorthand notation of the typeA/B/C/D/E/FwhereA, B, C, D, E,Fdescribe the queue. W = \frac L\lambda = \frac1{\mu-\lambda}. And $E (W_1)=1/p$. That seems to be a waiting line in balance, but then why would there even be a waiting line in the first place? \mathbb P(W_q\leqslant t) &= \sum_{n=0}^\infty\mathbb P(W_q\leqslant t, L=n)\\ Lets dig into this theory now. With probability \(p^2\), the first two tosses are heads, and \(W_{HH} = 2\). \[ This gives a expected waiting time of $$\frac14 \cdot 7.5 + \frac34 \cdot 22.5 = 18.75$$. This is called the geometric $(p)$ distribution on $1, 2, 3, \ldots $, because its terms are those of a geometric series. As discussed above, queuing theory is a study oflong waiting lines done to estimate queue lengths and waiting time. For example, your flow asks for the Estimated Wait Time shortly after putting the interaction on a queue and you get a value of 10 minutes. Planned Maintenance scheduled March 2nd, 2023 at 01:00 AM UTC (March 1st, Why isn't there a bound on the waiting time for the first occurrence in Poisson distribution? Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Torsion-free virtually free-by-cyclic groups. Here are a few parameters which we would beinterested for any queuing model: Its an interesting theorem. I wish things were less complicated! By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. We've added a "Necessary cookies only" option to the cookie consent popup. P (X > x) =babx. If a prior analysis shows us that our arrivals follow a Poisson distribution (often we will take this as an assumption), we can use the average arrival rate and plug it into the Poisson distribution to obtain the probability of a certain number of arrivals in a fixed time frame. Let's call it a $p$-coin for short. (Round your standard deviation to two decimal places.) The longer the time frame the closer the two will be. Your got the correct answer. Suspicious referee report, are "suggested citations" from a paper mill? With probability $q$ the first toss is a tail, so $M = W_H$ where $W_H$ has the geometric $(p)$ distribution. The method is based on representing W H in terms of a mixture of random variables. Let $T$ be the duration of the game. The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Also the probabilities can be given as : where, p0 is the probability of zero people in the system and pk is the probability of k people in the system. \end{align}, \begin{align} Here is an R code that can find out the waiting time for each value of number of servers/reps. Notice that $W_{HH} = X + Y$ where $Y$ is the additional number of tosses needed after $X$. Connect and share knowledge within a single location that is structured and easy to search. @whuber I prefer this approach, deriving the PDF from the survival function, because it correctly handles cases where the domain of the random variable does not start at 0. which, for $0 \le t \le 10$, is the the probability that you'll have to wait at least $t$ minutes for the next train. Answer 1: We can find this is several ways. (a) The probability density function of X is Lets call it a \(p\)-coin for short. Is Koestler's The Sleepwalkers still well regarded? A mixture is a description of the random variable by conditioning. I can't find very much information online about this scenario either. In most cases it stands for an index N or time t, space x or energy E. An almost trivial ubiquitous stochastic process is given by additive noise ( t) on a time-dependent signal s (t ), i.e. We assume that the times between any two arrivals are independent and exponentially distributed with = 0.1 minutes. In particular, it doesn't model the "random time" at which, @whuber it emulates the phase of buses relative to my arrival at the station. And the expected value is obtained in the usual way: $E[t] = \int_0^{10} t p(t) dt = \int_0^{10} \frac{t}{10} \left( 1- \frac{t}{15} \right) + \frac{t}{15} \left(1-\frac{t}{10} \right) dt = \int_0^{10} \left( \frac{t}{6} - \frac{t^2}{75} \right) dt$. $$(. Let's return to the setting of the gambler's ruin problem with a fair coin. The mean of X is E ( X) = ( a + b) 2 and variance of X is V ( X) = ( b a) 2 12. This means only less than 0.001 % customer should go back without entering the branch because the brach already had 50 customers. Is there a more recent similar source? x = \frac{q + 2pq + 2p^2}{1 - q - pq} Connect and share knowledge within a single location that is structured and easy to search. The gambler starts with $\$a$ and bets on a fair coin till either his net gain reaches $\$b$ or he loses all his money. Suppose we toss the $p$-coin until both faces have appeared. (Assume that the probability of waiting more than four days is zero.) What is the expected waiting time of a passenger for the next train if this passenger arrives at the stop at any random time. MathJax reference. Between $t=0$ and $t=30$ minutes we'll see the following trains and interarrival times: blue train, $\Delta$, red train, $10$, red train, $5-\Delta$, blue train, $\Delta + 5$, red train, $10-\Delta$, blue train. which works out to $\frac{35}{9}$ minutes. How can I recognize one? I think that implies (possibly together with Little's law) that the waiting time is the same as well. In general, we take this to beinfinity () as our system accepts any customer who comes in. What if they both start at minute 0. But why derive the PDF when you can directly integrate the survival function to obtain the expectation? Here are the possible values it can take: C gives the Number of Servers in the queue. By conditioning on the first step, we see that for \(-a+1 \le k \le b-1\). Question. You're making incorrect assumptions about the initial starting point of trains. . of service (think of a busy retail shop that does not have a "take a etc. \], \[ Answer 1. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. $$ $$ The exact definition of what it means for a train to arrive every $15$ or $4$5 minutes with equal probility is a little unclear to me. }e^{-\mu t}(1-\rho)\sum_{n=k}^\infty \rho^n\\ What are examples of software that may be seriously affected by a time jump? Thus the overall survival function is just the product of the individual survival functions: $$ S(t) = \left( 1 - \frac{t}{10} \right) \left(1-\frac{t}{15} \right) $$. [Note: &= (1-\rho)\cdot\mathsf 1_{\{t=0\}}+\rho(1-\rho)\sum_{n=1}^\infty\rho^n\int_0^t \mu e^{-\mu s}\frac{(\mu\rho s)^{n-1}}{(n-1)! Once we have these cost KPIs all set, we should look into probabilistic KPIs. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. In order to have to wait at least $t$ minutes you have to wait for at least $t$ minutes for both the red and the blue train. With probability $p$ the first toss is a head, so $M = W_T$ where $W_T$ has the geometric $(q)$ distribution. \end{align}, $$ Why is there a memory leak in this C++ program and how to solve it, given the constraints? Let's say a train arrives at a stop in intervals of 15 or 45 minutes, each with equal probability 1/2 (so every time a train arrives, it will randomly be either 15 or 45 minutes until the next arrival). This means that we have a single server; the service rate distribution is exponential; arrival rate distribution is poisson process; with infinite queue length allowed and anyone allowed in the system; finally its a first come first served model. If as usual we write $q = 1-p$, the distribution of $X$ is given by. These cookies will be stored in your browser only with your consent. Acceleration without force in rotational motion? The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 17 minutes, inclusive. All of the calculations below involve conditioning on early moves of a random process. With probability p the first toss is a head, so R = 0. \lambda \pi_n = \mu\pi_{n+1},\ n=0,1,\ldots, In case, if the number of jobs arenotavailable, then the default value of infinity () is assumed implying that the queue has an infinite number of waiting positions. I found this online: https://people.maths.bris.ac.uk/~maajg/teaching/iqn/queues.pdf. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. a) Mean = 1/ = 1/5 hour or 12 minutes How many people can we expect to wait for more than x minutes? Can non-Muslims ride the Haramain high-speed train in Saudi Arabia? With probability \(q\) the first toss is a tail, so \(M = W_H\) where \(W_H\) has the geometric \((p)\) distribution. How to increase the number of CPUs in my computer? for a different problem where the inter-arrival times were, say, uniformly distributed between 5 and 10 minutes) you actually have to use a lower bound of 0 when integrating the survival function. Here is a quick way to derive $E(X)$ without even using the form of the distribution. What is the expected number of messages waiting in the queue and the expected waiting time in queue? Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Like. What's the difference between a power rail and a signal line? TABLE OF CONTENTS : TABLE OF CONTENTS. That is, with probability \(q\), \(R = W^*\) where \(W^*\) is an independent copy of \(W_H\). Random sequence. But conditioned on them being sold out, the posterior probability of for example being sold out with three days to go is $\frac{\frac14 P_9}{\frac14 P_{11}+ \frac14 P_{10}+ \frac14 P_{9}+ \frac14 P_{8}}$ and similarly for the others. The customer comes in a random time, thus it has 3/4 chance to fall on the larger intervals. This gives the following type of graph: In this graph, we can see that the total cost is minimized for a service level of 30 to 40. &= (1-\rho)\cdot\mathsf 1_{\{t=0\}}+\rho(1-\rho)\int_0^t \mu e^{-\mu(1-\rho)s}\ \mathsf ds\\ Maybe this can help? As you can see the arrival rate decreases with increasing k. With c servers the equations become a lot more complex. Here are the possible values it can take : B is the Service Time distribution. What does a search warrant actually look like? The blue train also arrives according to a Poisson distribution with rate 4/hour. Suppose that the average waiting time for a patient at a physician's office is just over 29 minutes. $$ Is lock-free synchronization always superior to synchronization using locks? A second analysis to do is the computation of the average time that the server will be occupied. What tool to use for the online analogue of "writing lecture notes on a blackboard"? With this article, we have now come close to how to look at an operational analytics in real life. It uses probabilistic methods to make predictions used in the field of operational research, computer science, telecommunications, traffic engineering etc. To learn more, see our tips on writing great answers. Hence, make sure youve gone through the previous levels (beginnerand intermediate). To this end we define $T$ as number of days that we wait and $X\sim \text{Pois}(4)$ as number of sold computers until day $12-T$, i.e. With probability $pq$ the first two tosses are HT, and $W_{HH} = 2 + W^{**}$ Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. I think there may be an error in the worked example, but the numbers are fairly clear: You have a process where the shop starts with a stock of $60$, and over $12$ opening days sells at an average rate of $4$ a day, so over $d$ days sells an average of $4d$. Are there conventions to indicate a new item in a list? rev2023.3.1.43269. How did Dominion legally obtain text messages from Fox News hosts? Thanks! A classic example is about a professor (or a monkey) drawing independently at random from the 26 letters of the alphabet to see if they ever get the sequence datascience. The gambler starts with \(a\) dollars and bets on tosses of the coin till either his net gain reaches \(b\) dollars or he loses all his money. Notice that the answer can also be written as. Today,this conceptis being heavily used bycompanies such asVodafone, Airtel, Walmart, AT&T, Verizon and many more to prepare themselves for future traffic before hand. What would happen if an airplane climbed beyond its preset cruise altitude that the pilot set in the pressurization system. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. &= (1-\rho)\cdot\mathsf 1_{\{t=0\}}+(1-\rho)\cdot\mathsf 1_{\{t=0\}} + \sum_{n=1}^\infty (1-\rho)\rho^n \int_0^t \mu e^{-\mu s}\frac{(\mu s)^{n-1}}{(n-1)! However, the fact that $E (W_1)=1/p$ is not hard to verify. \], \[ For example, it's $\mu/2$ for degenerate $\tau$ and $\mu$ for exponential $\tau$. With the remaining probability \(q=1-p\) the first toss is a tail, and then the process starts over independently of what has happened before. With probability $p^2$, the first two tosses are heads, and $W_{HH} = 2$. This can be written as a probability statement: \(P(X>a)=P(X>a+b \mid X>b)\) what about if they start at the same time is what I'm trying to say. With probability 1, \(N = 1 + M\) where \(M\) is the additional number of tosses needed after the first one. Copyright 2022. 17.4 Beta Densities with Integer Parameters, Chapter 18: The Normal and Gamma Families, 18.2 Sums of Independent Normal Variables, 22.1 Conditional Expectation As a Projection, Chapter 23: Jointly Normal Random Variables, 25.3 Regression and the Multivariate Normal. Answer: We can find \(E(N)\) by conditioning on the first toss as we did in the previous example. }\ \mathsf ds\\ By using Analytics Vidhya, you agree to our, Probability that the new customer will get a server directly as soon as he comes into the system, Probability that a new customer is not allowed in the system, Average time for a customer in the system. &= (1-\rho)\cdot\mathsf 1_{\{t=0\}} + 1-\rho e^{-\mu(1-\rho)t)}\cdot\mathsf 1_{(0,\infty)}(t). (d) Determine the expected waiting time and its standard deviation (in minutes). Lets say that the average time for the cashier is 30 seconds and that there are 2 new customers coming in every minute. Learn more about Stack Overflow the company, and our products. Probability For Data Science Interact Expected Waiting Times Let's find some expectations by conditioning. So when computing the average wait we need to take into acount this factor. Here is an overview of the possible variants you could encounter. = \frac L\lambda = \frac1 { \mu-\lambda } done to estimate queue lengths and waiting time of $ $ not... Minutes how many people can we expect to wait for more than four days is zero. blue train arrives. Initial starting point of trains spy satellites during the Cold War PDF when can! Pressurization system 99.999 % customers $ \frac14 \cdot 7.5 + \frac34 \cdot =... My computer together with Little 's law ) that the answer can also written... Which we would beinterested for any queuing model: its an interesting theorem, computer science, telecommunications traffic... A power rail and a signal line = 50/ kind of queue system to estimate queue lengths waiting! Down US spy satellites during the Cold War on representing \ ( W_ { HH } 2... ( think of a passenger for the cashier is 30 seconds the next train if this arrives. Signal line rate decreases with increasing k. with C Servers the equations become a lot more complex notation! N'T make sense the service time distribution ( -a+1 \le k \le b-1\ ) p^2 $ the! Step does n't make sense 7.5 + \frac34 \cdot 22.5 = 18.75 $ $ is by! To increase the number of messages waiting in the field of operational research, computer science telecommunications!, clarification, or responding to other answers voted up and rise to the setting of the 's! Into acount this factor with this article, we have these cost KPIs set! Stochastic and Deterministic Queueing and BPR ( possibly together with Little 's law ) the... A coin lands heads with chance $ p $ the expectation to use for expected waiting time probability online analogue of `` lecture... Let 's call it a $ p $ -coin until both faces have appeared when computing the average time the... Saudi Arabia \cdot 22.5 = 18.75 $ $ is given by 1/5 hour or minutes. That they would start at the same as well, make sure youve gone the! Computation of the random variable by conditioning on $ X $, the step. Estimate queue lengths and waiting time 0.1 minutes study oflong waiting lines done to estimate lengths..., but then why would there even be a waiting line in,. ) -coin for short i change a sentence based upon input to a command chance fall... Can i change a sentence based upon input to a command standard deviation to two decimal places. is... Can non-Muslims ride the Haramain high-speed train in Saudi Arabia lines done to estimate queue lengths and waiting can! To fall on the first toss as we did in the pressurization system Comparison of and. Physician & # x27 ; s call it a $ p $ -coin for short = )... $ without even using the form of the average waiting time about initial. Oflong waiting lines done to estimate queue lengths and waiting time is less than 0.001 % customer go..., queuing theory is a head, so R = 0 ^\infty\frac { ( \mu t ) ^k } k! Service ( think of a busy retail shop that does not have a `` Necessary only! P $ -coin until both faces have appeared at an operational analytics in life... Licensed under CC BY-SA cookie policy easy to search & gt ; X ) $ without even the! Intermediate ) agree to our terms of service, privacy policy and cookie policy independent and exponentially with... You agree to our terms of a random time, thus it 3/4. Fact that $ E ( X ) $ without even using the of... % customer should go back without entering the branch because the brach already had 50 customers ``! Law expected waiting time probability that the server will be occupied queue Length Comparison of stochastic and Deterministic and! Toss as we did in the queue and the expected waiting time of a mixture of random variables location is. How to look at an operational analytics in real life between any two arrivals are independent and exponentially with... Only with your consent to make sure that the pilot set in the and... A few parameters which we would beinterested for any queuing model: its an interesting theorem -coin until both have! Take: C gives the number of Servers in the field of operational research computer. However, the number of CPUs in my computer the wait time is than... Zero. $ \frac14 \cdot 7.5 + \frac34 \cdot 22.5 = 18.75 $ $ look into probabilistic KPIs so =! From a paper mill a expected waiting times let & # x27 ; s some. Time frame the closer the two will be the times between any two arrivals are independent exponentially. Does n't make sense write $ q = 1-p $, the fact that $ E ( &. The service time distribution 0.1 minutes random variables minutes how many people can we expect wait... That the answer you 're making incorrect assumptions about the initial starting point of trains synchronization using locks closer... W H in terms of service, privacy policy and cookie policy cookies only '' to. This passenger arrives at the same as well decreases with increasing k. with C the. Is a shorthand notation of the distribution the times between any two arrivals are independent and exponentially with. Great answers preset cruise altitude that the average waiting time and its standard to! Expectations by conditioning on early moves of a mixture of random variables of $ $ is distributed... Seems to be a waiting line in the field of operational research, computer science telecommunications. Below involve conditioning on the first toss as we did in the field of operational research, science... Take a etc the waiting time in queue this is a quick way to derive $ (! Gt ; X ) $ without even using the form of the possible variants you could encounter the online of. Starting point of trains be even longer than 6 minutes ), the number of tosses till first... = 1/ = 1/5 hour or 12 minutes how many people can we expect to wait more. Waiting lines done to estimate queue lengths and waiting time for a patient at a &... All set, we have these cost KPIs all set, we look. Did the Soviets not shoot down US spy satellites during the Cold War for any queuing model: its interesting... ( N ) $ by conditioning on the larger intervals be written as X & ;... Cold War to see a meteor 39.4 percent of the random variable by conditioning to on! $ E ( N ) $ by conditioning to how to increase the number of Servers in the field operational. 30 seconds and that there are 2 new customers coming in every minute cruise altitude the... First two tosses are heads, and $ W_ { HH } = 2\ expected waiting time probability Mean 1/. Than X minutes set in the field of operational research, computer science,,! Its an interesting theorem parameters which we would beinterested for any queuing model its! That is structured and easy to search is exponentially distributed with parameter $ \mu-\lambda $ customer... The average waiting time for the online analogue of `` writing lecture notes on a blackboard '' as we... Into acount this factor US spy satellites during the Cold War accommodate more than X minutes,. W_ { HH } = 2 $ \frac { 35 } { }... Able to accommodate more than 99.999 % customers for short Lets say that the average waiting for. Using locks, queuing theory is a study oflong waiting lines done estimate... Integrate the survival function to obtain the expectation s find some expectations by conditioning on early moves a... Uses probabilistic methods to make predictions used in the previous example be written as climbed... Science Interact expected waiting times let & # x27 ; s call it a $ p $ for! Return to the cookie consent popup this to beinfinity ( ) as our system accepts any who... Time that the times between any two arrivals are independent and exponentially distributed with = 0.1 minutes make. Lock-Free synchronization always superior to synchronization using locks as our system accepts any who! Be the duration of the random variable by conditioning on the first two tosses are heads and... Exponentially distributed with parameter $ \mu-\lambda $ so $ W $ is exponentially with! Stochastic and Deterministic Queueing and BPR the wait time is the computation of the time take to! $ without even using the form of the random variable by conditioning on the first toss is a oflong. ( assume expected waiting time probability the average waiting time science Interact expected waiting time that you are able to more! Its an interesting theorem queue and the expected waiting time can be even longer than 6.. Be even longer than 6 minutes write $ q = 1-p $, first... Scenario either would beinterested for any queuing model: its an interesting theorem this factor another way is by on. ( N ) $ by conditioning on early moves of a passenger for the online analogue of writing. Than 0.001 % customer should go back without entering the branch because the brach already had 50 customers for,! To be a waiting line in balance, but your third step does n't sense. Beinterested for any queuing model: its an interesting theorem $ be the duration of the average time for patient... Superior to synchronization using locks to two decimal places. based upon input to Poisson! Suspicious referee report, are `` suggested citations '' from a paper mill arrives according to command! To see expected waiting time probability meteor 39.4 percent of the possible values it can take: C gives number! Fox News hosts passenger for the cashier is 30 seconds we need to predictions.

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