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 found this online: https://people.maths.bris.ac.uk/~maajg/teaching/iqn/queues.pdf. For definiteness suppose the first blue train arrives at time $t=0$. 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. }e^{-\mu t}(1-\rho)\sum_{n=k}^\infty \rho^n\\ The first waiting line we will dive into is the simplest waiting line. 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 expected waiting time for a single bus is half the expected waiting time for two buses and the variance for a single bus is half the variance of two buses. }e^{-\mu t}\rho^k\\ For example, if the first block of 11 ends in data and the next block starts with science, you will have seen the sequence datascience and stopped watching, even though both of those blocks would be called failures and the trials would continue. With the remaining probability $q$ the first toss is a tail, and then. Your got the correct answer. F represents the Queuing Discipline that is followed. There are alternatives, and we will see an example of this further on. To learn more, see our tips on writing great answers. I am probably wrong but assuming that each train's starting-time follows a uniform distribution, I would say that when arriving at the station at a random time the expected waiting time for: Suppose that red and blue trains arrive on time according to schedule, with the red schedule beginning $\Delta$ minutes after the blue schedule, for some $0\le\Delta<10$. rev2023.3.1.43269. I wish things were less complicated! MathJax reference. Let $X$ be the number of tosses of a $p$-coin till the first head appears. As discussed above, queuing theory is a study oflong waiting lines done to estimate queue lengths and waiting time. 2. All the examples below involve conditioning on early moves of a random process. 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. Let $L^a$ be the number of customers in the system immediately before an arrival, and $W_k$ the service time of the $k^{\mathrm{th}}$ customer. In this article, I will bring you closer to actual operations analytics usingQueuing theory. This means that the duration of service has an average, and a variation around that average that is given by the Exponential distribution formulas. 0. . Take a weighted coin, one whose probability of heads is p and whose probability of tails is therefore 1 p. Fix a positive integer k and continue to toss this coin until k heads in succession have resulted. Assume $\rho:=\frac\lambda\mu<1$. Like. One day you come into the store and there are no computers available. There is a blue train coming every 15 mins. And at a fast-food restaurant, you may encounter situations with multiple servers and a single waiting line. It uses probabilistic methods to make predictions used in the field of operational research, computer science, telecommunications, traffic engineering etc. Let's get back to the Waiting Paradox now. x = \frac{q + 2pq + 2p^2}{1 - q - pq} With probability $p^2$, the first two tosses are heads, and $W_{HH} = 2$. Lets say that the average time for the cashier is 30 seconds and that there are 2 new customers coming in every minute. $$ This minimizes an attacker's ability to eliminate the decoys using their age. a)If a sale just occurred, what is the expected waiting time until the next sale? Define a trial to be a "success" if those 11 letters are the sequence. With probability $q$, the first toss is a tail, so $W_{HH} = 1 + W^*$ where $W^*$ is an independent copy of $W_{HH}$. b)What is the probability that the next sale will happen in the next 6 minutes? I just don't know the mathematical approach for this problem and of course the exact true answer. Gamblers Ruin: Duration of the Game. A coin lands heads with chance $p$. Does Cast a Spell make you a spellcaster? }=1-\sum_{j=0}^{59} e^{-4d}\frac{(4d)^{j}}{j! With probability $q$ the first toss is a tail, so $M = W_H$ where $W_H$ has the geometric $(p)$ distribution. And $E (W_1)=1/p$. Hence, make sure youve gone through the previous levels (beginnerand intermediate). document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); How to Read and Write With CSV Files in Python:.. $$, \begin{align} &= (1-\rho)\cdot\mathsf 1_{\{t=0\}} + 1-\rho e^{-\mu(1-\rho)t)}\cdot\mathsf 1_{(0,\infty)}(t). With probability \(p\) the first toss is a head, so \(R = 0\). You need to make sure that you are able to accommodate more than 99.999% customers. The goal of waiting line models is to describe expected result KPIs of a waiting line system, without having to implement them for empirical observation. $$, $$ The customer comes in a random time, thus it has 3/4 chance to fall on the larger intervals. Answer 2: Another way is by conditioning on the toss after \(W_H\) where, as before, \(W_H\) is the number of tosses till the first head. Can I use a vintage derailleur adapter claw on a modern derailleur. This is a shorthand notation of the typeA/B/C/D/E/FwhereA, B, C, D, E,Fdescribe the queue. Since the exponential distribution is memoryless, your expected wait time is 6 minutes. Answer. I however do not seem to understand why and how it comes to these numbers. With this code we can compute/approximate the discrepancy between the expected number of patients and the inverse of the expected waiting time (1/16). Littles Resultthen states that these quantities will be related to each other as: This theorem comes in very handy to derive the waiting time given the queue length of the system. Queuing theory was first implemented in the beginning of 20th century to solve telephone calls congestion problems. In a 45 minute interval, you have to wait $45 \cdot \frac12 = 22.5$ minutes on average. Does exponential waiting time for an event imply that the event is Poisson-process? Dealing with hard questions during a software developer interview. $$. What if they both start at minute 0. Result KPIs for waiting lines can be for instance reduction of staffing costs or improvement of guest satisfaction. which yield the recurrence $\pi_n = \rho^n\pi_0$. Should I include the MIT licence of a library which I use from a CDN? It follows that $W = \sum_{k=1}^{L^a+1}W_k$. }e^{-\mu t}\rho^n(1-\rho) With probability $p$ the first toss is a head, so $M = W_T$ where $W_T$ has the geometric $(q)$ distribution. Since the schedule repeats every 30 minutes, conclude $\bar W_\Delta=\bar W_{\Delta+5}$, and it suffices to consider $0\le\Delta<5$. Suppose we toss the \(p\)-coin until both faces have appeared. The application of queuing theory is not limited to just call centre or banks or food joint queues. }\\ Expected waiting time. The best answers are voted up and rise to the top, Not the answer you're looking for? Lets understand these terms: Arrival rate is simply a resultof customer demand and companies donthave control on these. How did StorageTek STC 4305 use backing HDDs? \], \[ We've added a "Necessary cookies only" option to the cookie consent popup. Introduction. Now that we have discovered everything about the M/M/1 queue, we move on to some more complicated types of queues. Now you arrive at some random point on the line. \], \[ For example, waiting line models are very important for: Imagine a store with on average two people arriving in the waiting line every minute and two people leaving every minute as well. \lambda \pi_n = \mu\pi_{n+1},\ n=0,1,\ldots, Why was the nose gear of Concorde located so far aft? Total number of train arrivals Is also Poisson with rate 10/hour. Well now understandan important concept of queuing theory known as Kendalls notation & Little Theorem. In the second part, I will go in-depth into multiple specific queuing theory models, that can be used for specific waiting lines, as well as other applications of queueing theory. 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 $22\frac{1}{2}$. Is there a more recent similar source? Since the exponential mean is the reciprocal of the Poisson rate parameter. Look for example on a 24 hours time-line, 3/4 of it will be 45m intervals and only 1/4 of it will be the shorter 15m intervals. For example, suppose that an average of 30 customers per hour arrive at a store and the time between arrivals is . \end{align}$$ It only takes a minute to sign up. At what point of what we watch as the MCU movies the branching started? Conditional Expectation As a Projection, 24.3. Regression and the Bivariate Normal, 25.3. The various standard meanings associated with each of these letters are summarized below. Sometimes Expected number of units in the queue (E (m)) is requested, excluding customers being served, which is a different formula ( arrival rate multiplied by the average waiting time E(m) = E(w) ), and obviously results in a small number. All of the calculations below involve conditioning on early moves of a random process. $$ This phenomenon is called the waiting-time paradox [ 1, 2 ]. Lets dig into this theory now. Here is an R code that can find out the waiting time for each value of number of servers/reps. }.$ This gives $P_{11}$, $P_{10}$, $P_{9}$, $P_{8}$ as about $0.01253479$, $0.001879629$, $0.0001578351$, $0.000006406888$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. What would happen if an airplane climbed beyond its preset cruise altitude that the pilot set in the pressurization system? This should clarify what Borel meant when he said "improbable events never occur." Why? rev2023.3.1.43269. 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. But opting out of some of these cookies may affect your browsing experience. The most apparent applications of stochastic processes are time series of . Waiting till H A coin lands heads with chance $p$. Conditioning on $L^a$ yields Models with G can be interesting, but there are little formulas that have been identified for them. We also use third-party cookies that help us analyze and understand how you use this website. Acceleration without force in rotational motion? Answer 1. The number at the end is the number of servers from 1 to infinity. Suppose we do not know the order Here are the possible values it can take: C gives the Number of Servers in the queue. $$ E(x)= min a= min Previous question Next question 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. Define a "trial" to be 11 letters picked at random. Connect and share knowledge within a single location that is structured and easy to search. served is the most recent arrived. Why was the nose gear of Concorde located so far aft? X=0,1,2,. With probability \(pq\) the first two tosses are HT, and \(W_{HH} = 2 + W^{**}\) probability - Expected value of waiting time for the first of the two buses running every 10 and 15 minutes - Cross Validated Expected value of waiting time for the first of the two buses running every 10 and 15 minutes Asked 5 years, 4 months ago Modified 5 years, 4 months ago Viewed 7k times 20 I came across an interview question: Also, please do not post questions on more than one site you also posted this question on Cross Validated. $$. Find the probability that the second arrival in N_1 (t) occurs before the third arrival in N_2 (t). With probability $q$, the toss after $X$ is a tail, so $Y = 1 + W^*$ where $W^*$ is an independent copy of $W_{HH}$. Analytics Vidhya App for the Latest blog/Article, 15 Must Read Books for Entrepreneurs in Data Science, Big Data Architect Mumbai (5+ years of experience). With probability 1, at least one toss has to be made. E(X) = 1/ = 1/0.1= 10. minutes or that on average, buses arrive every 10 minutes. This takes into account the clarification of the the OP in a comment that the correct assumptions to take are that each train is on a fixed timetable independent of the other and of the traveller's arrival time, and that the phases of the two trains are uniformly distributed, $$ p(t) = (1-S(t))' = \frac{1}{10} \left( 1- \frac{t}{15} \right) + \frac{1}{15} \left(1-\frac{t}{10} \right) $$. How can the mass of an unstable composite particle become complex? c) To calculate for the probability that the elevator arrives in more than 1 minutes, we have the formula. Hence, it isnt any newly discovered concept. Can I use a vintage derailleur adapter claw on a modern derailleur. By additivity and averaging conditional expectations. Are there conventions to indicate a new item in a list? +1 I like this solution. All of the calculations below involve conditioning on early moves of a random process. An example of an Exponential distribution with an average waiting time of 1 minute can be seen here: For analysis of an M/M/1 queue we start with: From those inputs, using predefined formulas for the M/M/1 queue, we can find the KPIs for our waiting line model: It is often important to know whether our waiting line is stable (meaning that it will stay more or less the same size). They will, with probability 1, as you can see by overestimating the number of draws they have to make. Moreover, almost nobody acknowledges the fact that they had to make some such an interpretation of the question in order to obtain an answer. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. rev2023.3.1.43269. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. This means only less than 0.001 % customer should go back without entering the branch because the brach already had 50 customers. It is mandatory to procure user consent prior to running these cookies on your website. &= (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)! In real world, this is not the case. By conditioning on the first step, we see that for \(-a+1 \le k \le b-1\). Why is there a memory leak in this C++ program and how to solve it, given the constraints? If this is not given, then the default queuing discipline of FCFS is assumed. Let \(T\) be the duration of the game. $$, \begin{align} I remember reading this somewhere. Is Koestler's The Sleepwalkers still well regarded? 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). Another name for the domain is queuing theory. It has to be a positive integer. What the expected duration of the game? Could very old employee stock options still be accessible and viable? \mathbb P(W>t) &= \sum_{n=0}^\infty \mathbb P(W>t\mid L^a=n)\mathbb P(L^a=n)\\ Ackermann Function without Recursion or Stack. Bernoulli \((p)\) trials, the expected waiting time till the first success is \(1/p\). Random sequence. In a theme park ride, you generally have one line. The corresponding probabilities for $T=2$ is 0.001201, for $T=3$ it is 9.125e-05, and for $T=4$ it is 3.307e-06. So the average wait time is the area from $0$ to $30$ of an array of triangles, divided by $30$. Jordan's line about intimate parties in The Great Gatsby? Here is a quick way to derive $E(X)$ without even using the form of the distribution. 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. 5.Derive an analytical expression for the expected service time of a truck in this system. Thus the overall survival function is just the product of the individual survival functions: $$ S(t) = \left( 1 - \frac{t}{10} \right) \left(1-\frac{t}{15} \right) $$. $$ L = \mathbb E[\pi] = \sum_{n=1}^\infty n\pi_n = \sum_{n=1}^\infty n\rho^n(1-\rho) = \frac\rho{1-\rho}. The following example shows how likely it is for each number of clients arriving if the arrival rate is 1 per time and the arrivals follow a Poisson distribution. Suspicious referee report, are "suggested citations" from a paper mill? We can expect to wait six minutes or less to see a meteor 39.4 percent of the time. Let \(E_k(T)\) denote the expected duration of the game given that the gambler starts with a net gain of \(k\) dollars. An interesting business-oriented approach to modeling waiting lines is to analyze at what point your waiting time starts to have a negative financial impact on your sales. &= (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)! Suppose we toss the $p$-coin until both faces have appeared. There is nothing special about the sequence datascience. Think of what all factors can we be interested in? Even though we could serve more clients at a service level of 50, this does not weigh up to the cost of staffing. The best answers are voted up and rise to the top, Not the answer you're looking for? What does a search warrant actually look like? Possible values are : The simplest member of queue model is M/M/1///FCFS. Let $X(t)$ be the number of customers in the system at time $t$, $\lambda$ the arrival rate, and $\mu$ the service rate. To learn more, see our tips on writing great answers. Some analyses have been done on G queues but I prefer to focus on more practical and intuitive models with combinations of M and D. Lets have a look at three well known queues: An example of this is a waiting line in a fast-food drive-through, where everyone stands in the same line, and will be served by one of the multiple servers, as long as arrivals are Poisson and service time is Exponentially distributed. In the supermarket, you have multiple cashiers with each their own waiting line. And the expected value is obtained in the usual way: $E[t] = \int_0^{10} t p(t) dt = \int_0^{10} \frac{t}{10} \left( 1- \frac{t}{15} \right) + \frac{t}{15} \left(1-\frac{t}{10} \right) dt = \int_0^{10} \left( \frac{t}{6} - \frac{t^2}{75} \right) dt$. The time between train arrivals is exponential with mean 6 minutes. $$ To visualize the distribution of waiting times, we can once again run a (simulated) experiment. probability probability-theory operations-research queueing-theory Share Cite Follow edited Nov 6, 2019 at 5:59 asked Nov 5, 2019 at 18:15 user720606 Once every fourteen days the store's stock is replenished with 60 computers. With probability 1, at least one toss has to be made. We can find this is several ways. A Medium publication sharing concepts, ideas and codes. For example, it's $\mu/2$ for degenerate $\tau$ and $\mu$ for exponential $\tau$. The probability that total waiting time is between 3 and 8 minutes is P(3 Y 8) = F(8)F(3) = . 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. \begin{align} After reading this article, you should have an understanding of different waiting line models that are well-known analytically. We can find $E(N)$ by conditioning on the first toss as we did in the previous example. Can trains not arrive at minute 0 and at minute 60? E_{-a}(T) = 0 = E_{a+b}(T) Answer. But some assumption like this is necessary. How to react to a students panic attack in an oral exam? Waiting line models need arrival, waiting and service. Quick way to derive $ E ( X ) = 1/ = 1/0.1= 10. minutes or that on,. \Begin { align } I remember reading this article, you should have an understanding of waiting. & Little Theorem and then more, see our tips on writing great answers an unstable composite particle become?. Or improvement of guest satisfaction consent prior to running these cookies may affect your browsing.... Are able to accommodate more than 1 minutes, we have the formula what Borel meant when he &... '' to be made { n+1 }, \ [ we 've added a `` trial '' be... The third expected waiting time probability in N_2 ( t ) occurs before the third arrival N_2. Only less than 0.001 % customer should go back without entering the branch because the brach already had 50.! Coming in every minute average time for an event imply that the event is Poisson-process the line copy and this. Minutes, we can once again run a ( simulated ) experiment this not! I just do n't know the mathematical approach for this problem and of course the exact true answer {! $ the first blue train arrives at time $ t=0 $ does not weigh up to the cost of.! Of these cookies on your website M/M/1 queue, we see that \! Of stochastic processes are time series of to learn more, see our tips on writing answers. Than 0.001 % customer should go back without entering the branch because the brach already had 50.... \Mu/2 $ expected waiting time probability exponential $ \tau $ and $ \mu $ for degenerate $ \tau.. Concorde located so far aft how it comes to these numbers 50 customers to RSS. Think of what all factors can we be interested in exponential mean is the expected waiting time till first. Had 50 customers exponential waiting time \mu\pi_ { n+1 }, \ n=0,1, \ldots why. Waiting-Time Paradox [ 1, as you can see by overestimating the number servers/reps! Mandatory to procure user consent prior to running these cookies on your website events never occur. & quot ; events! Professionals in related fields, but there are Little formulas that have been identified for.! The waiting Paradox now W_k $ could very old employee stock options still be accessible and viable 2. Model is M/M/1///FCFS X $ be the duration of the time form of the Poisson parameter! Analytics usingQueuing theory cruise expected waiting time probability that the second arrival in N_2 ( t ) traffic engineering etc takes... Fdescribe the queue 30 seconds and that there are Little formulas that been... True answer letters picked at random be interesting, but there are no computers available we. Professionals in related fields M/M/1 queue, we move on to some complicated! This RSS feed, copy and paste this URL into your RSS.... Line models that are well-known analytically number at the end is the reciprocal of the typeA/B/C/D/E/FwhereA b. Customers coming in every minute is also Poisson with rate 10/hour it comes to these numbers real,! Also Poisson with rate 10/hour what point of what we watch as MCU. Understand why and how it comes to these numbers discovered everything about the M/M/1,... $ \mu $ for degenerate $ \tau $ ) trials expected waiting time probability the expected waiting time for value. Movies the branching started operational research, computer science, telecommunications, traffic etc.: arrival rate is simply a resultof customer demand and companies donthave control these! The exact true answer exponential $ \tau $ and $ \mu $ for exponential $ \tau $ and $ $... 'Ve added a `` Necessary cookies only '' option to the top, not answer... Is simply a resultof customer demand and companies donthave control on these ; s back. Option to the cookie consent popup understand how you use this website 2 ] truck in this C++ program how. Ride, you generally have one line coming in every minute remember reading this somewhere, as can... Stack Exchange is a question and answer site for people studying math at any level and in., computer science, telecommunications, traffic engineering etc very old employee stock options still be accessible and viable,! Suppose we toss the \ ( 1/p\ ), computer science, telecommunications, traffic etc! About the M/M/1 queue, we can once again run a ( simulated ) experiment D,,... Everything about the M/M/1 queue, we have the formula I use from a CDN number of servers/reps to. Value of number of train arrivals is also Poisson with rate 10/hour, D, E Fdescribe... Of FCFS is assumed stochastic processes are time series of 0\ ) of further... Known as Kendalls notation & Little Theorem is 30 seconds and that there are no computers available first implemented the! Expected waiting time for the probability that the elevator arrives in more than 1 minutes, can... ) what is the expected service time of a truck in this system expected waiting time probability lands heads with chance $ $... That we have the formula this C++ program and how it comes to these.!, \begin { align } After reading this article, you may encounter situations with multiple servers and single. With rate 10/hour what point of what we watch as the MCU movies the branching?. Elevator arrives in more than 99.999 % customers to eliminate the decoys using their age the of. Is simply a resultof customer demand and companies donthave control on these does exponential time. Paradox [ 1, as you can see by overestimating the number of tosses a! Way to derive $ E ( N ) $ without even using form! Brach already had 50 customers remember reading this article, I will bring you to... Your browsing experience since the exponential distribution is memoryless, your expected wait time is 6 minutes first in. $ q $ the first success is \ ( p\ ) -coin until both faces have appeared faces! Of an unstable composite particle become complex our tips on writing great answers typeA/B/C/D/E/FwhereA b! ( p ) \ ) trials, the expected waiting time for an event that. Publication sharing concepts, ideas and codes the Poisson rate parameter out the waiting time for an event that... Train arrives at time $ t=0 $ not weigh up to the waiting now. Supermarket, you may encounter situations with multiple servers and a single location that is structured and to! Your browsing experience theory was first implemented in the supermarket, you multiple... To react to a students panic attack in an oral exam \begin { align } $,. Are Little formulas that have been identified for them $ to visualize the distribution of waiting times we. Third-Party cookies that help us analyze and understand how you use this website this! The distribution theory was first implemented in the beginning of 20th century to solve it, given constraints. Is also Poisson with rate 10/hour 've added a `` trial '' be... A head, so \ ( R = 0\ ) of an unstable composite particle become?!, waiting and service Little formulas that have been identified for them, it 's $ $. Is mandatory to procure user consent prior to running these cookies may affect your browsing experience distribution. We have the formula lines can be interesting, but there are alternatives, and then exponential distribution memoryless... For exponential $ \tau $ know the mathematical approach for this problem and of course exact! However do not seem to understand why and how it comes to these numbers the Gatsby... ( R = 0\ ) in this system actual operations analytics usingQueuing theory, but are., this is not given, then the default queuing discipline of FCFS is assumed so (... Kendalls notation & Little Theorem the M/M/1 queue, we see that for \ ( 1/p\ ) the time a... Notation of the Poisson rate parameter ideas and codes the cost of staffing are Little formulas that have been for! The mathematical approach for this problem and of course the exact true answer cookies only '' to... Beginning of 20th century to solve telephone calls congestion problems what Borel meant when said... Beginning of 20th century to solve telephone calls congestion problems we toss the \ ( 1/p\ ) an. Library which I use from a CDN location that is structured and easy to.... $ this phenomenon is called the waiting-time Paradox [ 1, as you can see by overestimating the of. Subscribe to this RSS feed, copy and paste this URL into your RSS reader said & quot ; events... Series of mass of an unstable composite particle become complex prior to running cookies. Cookie consent popup probability 1, as you can see by overestimating the number of draws they have make. To running these cookies may affect your browsing experience though we could serve more clients at a fast-food,. Can be interesting, but there are no computers available we see that for \ ( ( p ) )... Known as Kendalls notation & Little Theorem no computers available suppose that an average of customers! \Le k \le b-1\ ) what Borel meant when he said & quot ; improbable events never &! `` success '' if those 11 letters picked at random beyond its cruise... A `` trial '' to be 11 letters picked at random be 11 letters picked at random should an! Beginning of 20th century to solve telephone calls congestion problems eliminate the decoys using their age telephone... To make predictions used in the great Gatsby most apparent applications of processes! Lengths and waiting time understanding of different waiting line youve gone through the levels... Six minutes or that on average, buses arrive every 10 minutes developer interview to infinity then the default discipline!

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