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

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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 ...
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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 ...
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34 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 ...
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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 ...
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24 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)).
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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. ...
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64 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. ...
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29 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 ...
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26 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. ...
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38 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 ...
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17 views

Optimal scheduling policy for two server model

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 ...
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39 views

m/m/1 Queuing Theory - Average Delay/Utilization Graph confusion

I'm currently coding a single server single queue m/m/1 simulation with Lambda as arrival events rate (Poisson) Omega as service rate (Exponential) and I'm having problem understanding how to ...
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parameters of queuing theory

Consider a queuing system that follows the model M / M / 1 wherein the average time between consecutive arrivals is 12 secons and the average service time is 3 seconds. ...
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28 views

Can I think of both arrival times and service times in a Markov chain as Poisson processes?

According to the Wikipedia article about M/M/1 queues, the rate at which new jobs arrive is a Poisson process with parameter $\lambda$, and the rate at which the jobs are finished is an exponential ...
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36 views

Applying Generating Function Approach to a $M/E_r/1$ queue

(This question is about Exercise 27 on page 55 from these lecture notes.) We consider a $M/E_r/1$ queue with arrival rate $\lambda$ and mean service time $r/\mu$. We let ...
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Formula for Pipelining

Can someone help me come up with a formula to describe the time it takes for a process to take when using pipelining? For example, suppose I have to send a message from ...
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35 views

For a M/M/1/K System, why is utilization = $\rho=\lambda T_s$

Before I ask my question these are the symbols I used: $w$ = Customers waiting in queue $\mu$ = Service rate $T_s = 1/\mu$ = Average service time $\lambda$ = Average arrival time $\rho$ = Server ...
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31 views

Multiclass Markov process

There are two car M/M/1 queues Q1 and Q2. Arrival rate of Red car and Green car in Q1 is $\lambda_{1R}$ and $\lambda_{1G}$ respectively. Similarly arrival rate of red car and green car in Q2 is ...
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30 views

Funny queuing model probability distribution question

I like toilet jokes, so I am interested in mathematical Problems like follows: A Boy has a bladder with volume $V$. In his bladder it is inflowing the amount of urine (in cm³/sec) denoted by $\mu$; ...
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25 views

Survival function of birth-death process

There is a linear birth-death process with $N$ states + an absorbing state $0$, with $$\Pr[X_{t+1}=0|X_{t}=0]=1, \\ \Pr[X_{t+1}=i+1|X_{t}=i]=\Pr[X_{t+1}=i-1|X_{t}=i]=q_i, i\in [1..N-1],$$ and ...
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37 views

One queue vs multiple queues: how to calculate mean and variance?

I am having trouble understanding the solution to a queueing problem. You are at an amusement park. There are two ticketing areas. The first one has 1 ticketing machine and a 10-person line. The ...
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38 views

Calculating optimal priorities in M/M/c queueing system

This isn't a "solvable" question per-se, it's more a question of whether this can be solved, or modeled, and how I would do it. It's basically a more complex M/M/c system. The scenario: Customer ...
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How to calculate the optimal number of serverss in a M/D/c queue?

Assuming that the arrivals occur according to a Poisson process and that the time to process each arrival is fixed, how can the optimal number of servers 'c' be calculated such that the waiting time ...
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24 views

mean and std deviation from little's law

I have a service node with a queue The arrival rate in this service node is exponential(0.4) The service time is exponential(0.2) Running a simulation i've calculated the average population of the ...
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How do I find the average waiting time and standard error of the kth person in queue?

Given the sample average wait time for the 1st person in queue is Wq, and the sample variance of the wait time for the 1st person in queue is Vq, how do you calculate the average wait time for the Kth ...
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Distribution of the number of customers in a system

Consider the following scenario. Groups of potential customers arrive according to a Poisson process with a rate of 5 groups per hour at a single server service station. When a group arrives, with ...
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In M/M/1 Markov process, why must entering and leaving the zero state be equal?

According to the image below, which I snipped from this article, the rate of leaving State 0 and the rate of arriving into ...
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M/M/1 Queue Under Heavy/Light Loads

Given, $\rho \approx 1$ (heavy-load), for the open M/M/1 queue, can we consider the sequence of response times an approximate renewal process? That is, the sequence of measured response times from a ...
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Dynamics of birth-death process with discouraged arrivals (alternatively, M/M/1 queue with balking customers)

Take a continuous-time birth-death process, where $k \in \{0,1,\ldots\}$ is the count and the arrival rate of death is $\mu \geq 0$ for $k = 1, 2, \ldots$ the arrival rate of births is $\alpha_k ...
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56 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
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81 views

M/M/1/FIFO queue with multiple streams, delay of each stream?

Suppose that you have an M/M/1/FIFO queue where jobs arrive with two poisson arrival processes P1, P2. The service rate of the queue is $\mu$. The arrival rate of P1 and P2 is $\lambda_1$ and ...
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80 views

Countable state Markov chain with multiple transitions

I'm searching for hints on how to analyze the following Markov chain. I can solve for the steady state probabilities numerically by using a finite transition matrix. However, I would like to have an ...
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Queuing Theory: Computing Delay Probability M/M/C

The delay probability is : So let's assume that : λ = 2/3; μ = 1/3; c = 3; How do they arrive by having a queuing delay : Πw = 0.444 ? I don't know if I ...
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Invariant Distribution of Two Dependent $\cdot/M/\infty$ Queues Running in Parallel

This is in preparation for an exam I have coming up. We have two $\cdot / M / \infty$ queues with external arrivals occurring according to a Poisson Process of rate $\lambda$. Service occurs with ...
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Queueing model for web server

Suppose i have a single web server machine on which clients send requests in form of poisson distribution and all the clients have different service times,now i am confused whether i will take this ...
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41 views

Expected number of lines in use in call centre (markov process: queuing theory)

Suppose we have a call centre with infinitely many lines to be able to call to. Calls come in a rate of $\lambda$ and customers are served with rate $\mu$. It is easy to see that the $Q$-matris looks ...
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49 views

A queueing model issue.

I am very beginner in Queueing Theory and I am learning in my own. I am struggling in the following situation. Suppose in a service center if a job arrives it will immediately being processed if a ...
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63 views

M/M/1 and M/M/m queueing models for a client-server-database system

I have designed a system and have to do now analysis using queueing models with which I have problems. First, I will briefly explain my system. I have a client-server-database system. There are N ...
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32 views

Pollaczek–Khinchine formula for claims with expotential distribution - derivation

I am trying to understand ruin probability formula using Pollaczek–Khinchine formula described here: http://en.wikipedia.org/wiki/Ruin_theory $$\psi(x)=\left(1-\frac{\lambda ...
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Stationary probability in an M/M/$1$ queue with a lazy server

Customers arrive to a single server queue according to a Poisson process with rate $\lambda$. Each customer requires Exponential($\mu$) service time. In the beginning when there are $0$ ...
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What is the probability of a M/M/1 queue having less than 5 customers?

(Not homework) I'm doing some studying for an upcoming final, which includes material on queueing theory (the class is on design/analysis of computer networks, not queueing theory). I've worked ...
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63 views

Queue-length and waiting time of M/D/1-queue

I studied the M/G/1 queue by myself. Now as an application, I considered the M/D/1- queue. I know that the results can be find in the internet, but I haven't seen any calculations. Let $Q$ be an ...
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How many staff should this employer hire given this set of customer queue and service speed?

Suppose James owns a restaurant and gets on average X customers per minute and each takes Y minutes to be served. What is the minimum number of staff to ensure that 99% of customers are served in ...
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Proportion of arrivals taking a particular path in a Routing Matrix

I have a routing matrix with Node-0 being the source/sink (outside world) and there are service Nodes 1,2..k in the system. The matrix has entries R_ij = Probability of an arrival at Node-i moving to ...
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Analysis of shared server queueing system

To be able to analyse the processes in an organizational department, I had the idea that the processes can be modelled as a queueing system. Now suppose the service times are exponentially ...
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From an M/G/1-queue to the M/M/1-queue

Suppose I have an $M/G/1$-queue $Q$. Then I have an embedded Markov chain $Q(D)$ and the following theorem: If $\rho=\lambda \mathbb{E}[S]<1$, then $Q(D)$ is ergodic with a unique stationary ...
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M/M/1 queue, probability of time spending in queue..

Let $W$ be the time $nth$ customer spends in the queue when $n$ go to $\infty$. How do we write down the formula for $P[W \le t | N = k]$ ? where $N$ is the number of customer in the system.
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time for n or more than n customers in a M/M/1 queue system

The question asks for how much time (given an 8hr working day) in a day are there 2 or more customers in the system. If I am to calculate probability for 2 customers in the system and probability for ...
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63 views

Expected Service Times for truncated exponential

I'm trying to solve a problem where all arriving items (arrival exponential $\lambda = 1/5$) are divided into into groups, those who are served within 5 units of time and those who have their service ...
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Need guidance on a Queuing problem

I can't really go into specifics, I'm more just looking for terms that I can research to get on the right track. Classes of model/processes etc. A close analogy to my problem: I need to optimally ...