# Tagged Questions

Queueing theory is the mathematical study of waiting lines, or queues.

15 views

### Why multiple Poisson streams taken as sum of IID rather than joint PDF?

I have a very basic doubt regarding queueing theory. We argue that merging of more than one IID Poisson streams (say $i$ streams) with respective parameters $\lambda_i$ is also Poisson with parameter ...
21 views

### Wait time in queue for 2 server system (exponential process)

A customer has to be server 1 before being served by server 2. Service times are exponential with rates $\mu_1$ and $\mu_2$ respectively. After being done at server 1, you wait at that station till ...
16 views

### Transition Probability of M/M/1 Queue given any constant observing period T

I am trying to find the transition probability for $M$/$M$/$1$ queue given any constant observing period $T$ (if $T$ go to infinity the transition matrix will degenerate to a matrix with identical ...
24 views

### model with markov chain

Suppose to have the following situation: At a bar at each time unit arrives a certain number of customer with probabilities $p_1,p_2,...,p_n$. In the bar there are 3 bartenders so 3 customer can be ...
35 views

### Probability Applications

Trucks arrive at a specific petrol station, occur at a mean of 2 cars per minute but the probability distribution is not known. The normal staff for this petrol station can meet order requirements ...
23 views

### Average customers/hour equivalent to arrival rate?

For a Basic Single Server problem, M/M/1, I'm given the average customers/hour at clerks desk. The solution takes that figure as λ. That doesn't make sense, that figure is essentially how many ...
35 views

### Inversion of Laplace transforms - simplifying the Bromwich integral

I have trouble following the derivation of equation $(2)$ in this paper. The authors define the Laplace transform of a real-valued function $f(t)$ of a positive real variable $t$ as ...
23 views

### Understanding the Fluid limit model in Queueing Theory

I am self-studying the fluid limit model in queuing theory. The basic setting is that: let $Q(t)$ be the total number of customers in the system. Consider the simplest case of M/M/1 queuing system. ...
16 views

### Poisson arrivals, poisson departures with rate proportional to queue length

Suppose arrivals to a queue (/collection, more accurately) follow $Poisson(\lambda_a)$. In other words, the arrival rate is constant $\lambda_a$. Departures, however, at time $t$ go at rate ...
10 views

### Pay off of adding new service station

I have to solve the following problem: In a factory they do reperations of a certain type of machine. The machines that need reparations arrives at the factory with a frequency of 4 every hour, such ...
36 views

### interarrival time

I am trying to model the arrival rate (delay between the arrivals) of the cars in my city. I have some real data with the resolution of one minute. For example, Number of counted cars at position x at ...
47 views

### What is the probability of never reaching a state in Markov chain?

The markov chain consists of states $a_1$,...$a_k$. The initial state is $a_1$. For the state $a_{i}$ the next state can either be $a_{i+1}$ with probability p, or the initial state $a_1$ with ...
24 views

### What is the name of this problem about queuing processes?

The problem is: suppose we have $p$ processes labeled $1, \ldots, p$, with runtimes $(t_1, \ldots, t_p) \in \mathbb{R}_{++}^p$. In what order should the processes be set up to minimize overall runtime ...
40 views

### Combined arrival rate

Let us suppose a scenario with two clients, $a$ and $b$, each one generating load at rate $\lambda_a$ and $\lambda_b$, respectively. The server receives the requests from both clients. What will be ...
19 views

### Is there a Little law for a network of two connected queues?

From Patterson et al' Computer Organization and Design: Throughput and Response Time Do the following changes to a computer system increase throughput, decrease response time, or both? ...
30 views

### Survey on large deviation bounds of queuing delay in CSMA scheduling

I am trying to do some literature survey on the theoretical guarantees in uplink scheduling algorithms. I found there exist a series of papers from UIUC and UC Berkeley by L.Jiang, J. Walrand, R. ...
65 views

### What is the capacity of Earth? Is the Earth stable?

I'm learning queueing theory and just finished Little's Law and Utilization. If the Earth is interpreted as a system that provides a service for its customers, is it unstable? Let the customers be ...
104 views

### Probability distribution of number of waiting customers in front of a counter [closed]

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
43 views

### Confusion regarding Burke's theorem

Arrivals occur at rate $\lambda$ according to a Poisson process the service time have an exponential distribution with parameter $1/\mu$ in an M/M/1 queue, where $\mu$ is the mean service rate where ...
29 views

### Effective inter-arrival time converge to mean

I am fairly new to statistics and just recently encountered queueing theory. I have programmed a simulation for a $M/M/1$ queue in which I specify the inter-arrival times and service times. I input ...
144 views

### From one-dimensional to two-dimensional Markov chains

I have a $M/M/1$ queueing system that is described below: There are two types of customers in the system with different arrival rates, $\lambda_{sg}$ and $\lambda_{sb}$. Service rate is $\mu$. Type ...
55 views

### Final rank, given initial rank and a probability distribution

I got an interesting problem today, though I could not find a closed form solution to it. Imagine a setting where we have people submitting solutions, which are ranked. Given some initial rank of an ...
62 views

### Probability that a birth--death process crosses level $n$ in $(0,T)$

This question is inspired by this question. Jobs arriving according to a Poisson process with rate $\lambda$. Jobs stay in the system for a fixed amount of time $d$ and depart thereafter. Let $X(t)$ ...
60 views

### How to compute the arrival rate

I have a data set of interarrivals . I need to compute the arrival rate. Should I compute the mean interarrival and then inverse it to get the arrival rate ? or should I inverse all the ...
55 views

### Queue theory - M/D/k - Probability of never having a queue before a time T

This is probably a known result, but I couldn't find any resource pointing directly to the issue I'm trying to solve. Suppose you start a logistic mission that needs that during its time $T_m$ a ...
85 views

### M/M/1 vs G/G/1 vs G/M/1

I am using queuing theory to model a router. I have a model that assumes Poisson traffic and I need to modify it as my actual traffic is not Poisson I want to ask what's the main difference between ...
16 views

### Customers and Anti-Customer Queueing Problem: What is the Customer delete probability

Hello may ask for your help? First the setting: I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...
27 views

### M/M/1 Queueing theory with change in rates

Consider a single-server exponential system in which customers arrive at a rate $\lambda$ and have a regular service rate $\mu$. When a customers arrives and the system is busy, the customer joins the ...
27 views

### Probability of a message being processed in a given interval

Let's say I have a queue and a server that connects to the queue in an interval basis and process all the available messages. The server connects, tries to get a message, if there is one then server ...
39 views

### Expectation of the time difference between starting times in queueing theory

Consider 2 independent, parallel $M/M/1$ queues $Q_1, Q_2$ with identical arrival rate $\lambda$ (corresponding to an exponential random variable $A \sim \text{Exp}(\lambda)$) and service rate $\mu$ ...
31 views

### A type of continuous time Markov process

I am looking for a stochastic process model with the following features. It is a continuous time Markov process---modelling, if you like, the evolution of a population. New arrivals are added to ...
124 views

### Jackson's theorem to optimize mean queue length of a traffic model

I am working on traffic signals for a city transport system. I modeled the city transport using a queuing network as shown in the following image Arrival rate of "A" cars from outside is S1 and ...
40 views

### Laplace-Stieltjes transform M/G/1-queue

Two types of jobs arrive at a machine. Type $1$ jobs arrive according to a Poisson process with rate 45 per hour and need an exponential service time with mean $1/2$ minute. Type $2$ jobs arrive ...
45 views

### Distribution of the number of customers

We have the following situation Passengers are brought with small vans from the airport to hotels nearby. At one of those hotels on average $6$ vans per hour arrive according to a Poisson process. ...
128 views

### Distribution of server utilisations in an M/M/c queuing model with an unusual dispatching discipline

I'm studying an M/M/c queuing model with an unusual (?) dispatching discipline: Servers are numbered 1...c The servers have an identical mean service time, exponentially distributed (as usual), ...
60 views

### 2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
63 views

### Problem with Little's law

I am estimating number of threads required by my server to execute clients requests efficiently and initially I starts 4 threads on the server.Request arrival rate on my server is 4 request/sec and ...
96 views

### Time between arrivals Distribution

I am simulating a hair parlor queue with m number of queues and 3 different types of services (queues). I was doing the time between arrival with a uniform distribution with a min value and a max ...
54 views

### Why Little law is not fit for web request arrival rate scenario?

I am estimating number of threads required by my Server to execute clients requests efficiently and initially i starts 4 threads on the server.Request arrival rate on my server is 4 request/sec and ...
56 views

### Question about $M/GI/ \infty$ queue

Consider an $M/GI/ \infty$ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...
89 views

### Can You Help Me With This Continuous Markov Chain Question?

Consider 2 machines, both of which have an exponential lifetime with mean $\frac{1}{\lambda}$. There is a single repairman that can service machines at an exponential rate $\mu$. Set up the ...
217 views

### Interarrival time for Counting process?

Messages arrive at an interactive message center according to a counting process with the average inter-arrival time of 15 seconds. Choosing a frame size of 5 seconds, compute the probability that ...
38 views

### M/G/1 queueing problem

I need to prove that in the M/G/1 queueing system with Poisson arrivals with parameter lambda and exponential service time with parameter mu, that q_k = (lambda/(lambda+mu))^k (mu/(lambda+mu)).
69 views

### Compute the standard deviation of the monthly cost due to blackouts

Network blackouts occur at an average rate of 5 blackouts per month. Assuming a suitable continuous-time counting process, a. Compute the probability of more than 3 blackouts during a given month. ...
210 views

### Determine the probability that the time interval between successive job arrivals is :

Consider a server with Poisson job-arrival stream at an average rate of 60 per hour. Determine the probability that the time interval between successive job arrivals is a. longer than 4 min b. ...
32 views

### queuing problem, related to marriage algorithm

Say we have an nxn matrix and for every entry a_{ij}, it equals 1 if flight j starts after flight i ends. Otherwise it is 0. Suppose the largest matching contains M marriages (i.e. 1's in nxn matrix ...
51 views

### Markov Process - formulate a Markov chain model for this system ( what is q(i,j)?)

Potential customers arrive at a full-service, two-pump gas station according to a Poisson process at a rate of 40 cars per hour. There are two service attendants to help customers, one for each pump. ...
40 views

### Modelling a continious-time queue which behaves differently when there are more or less people being served.

For my research I am trying to model a continuous-time queue which behaves differently when there are more or less people being served. The arrival rate in the queue is constant, however the departure ...
Suppose there are two servers with exponential arrival rates $\mu_{1}$ and $\mu_2$ such that $\mu_1 > \mu_2$. These two servers have a shared infinite buffer, where there is independent Poisson ...