The M/M/m/m Server Loss System: Consider the queuing system given by the following state- transition diagram.
Each arriving customer is given a private server, but there is a maximum of m servers available. If a customer arrives when all m servers are busy, the customer is denied service and is turned away. The arrival rate is Poisson with parameter λ and the service rate is kμ with 1 ≤ k ≤ m as shown. Use the results obtained in Set 11(see the attached notes) to prove that
(see the attached notes):
ks m k! k 0 k>m k m 1 k! k 0 | S. l
M/M/1/K Queuing System Balance equation at state 0: A no-u Balance equation at state n: (u +) x, = ^ «,-1+iT Balance equation at state K: lnK-1= We need to solve the above K+ 1 equations to express r, in terms of and We can show that, for p
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The M/M/m/m Server Loss System: Consider the queuing system given by the following state- transition diagram. Each arri...
The M/M/1 and M/M/1/K queuing system: Consider the M/M/1 and M/M/1/K queuing systems [see in class notes]. For the M/M/1...
The M/M/1 and M/M/1/K queuing system: Consider the M/M/1 and
M/M/1/K queuing systems [see in class notes]. For the M/M/1/K
system show that, for ρ < 1,
in class notes:
p" (1-p) п-0, 1,2, ..., К-1; р-— 1-р*а п, K+1 и N- Р_(К+1)pku К-+1 1-р*а 1-р M/M/1 Queuing System with Finite Capacity (M/M/1/K) Systems have a finite capacity for serving customers. The M/M/1 queuing system capable of supporting up to K customers is called an M/M/1/K queuing system. Arrivals at...
Consider the M/M/1/GD/∞/∞ queuing system where λ and μ are the arrival and server rate, respectively....
Consider the M/M/1/GD/∞/∞ queuing system where λ and μ are the arrival and server rate, respectively. Suppose customers arrive according to a rate given by λ = 12 customers per hour and that service time is exponential with a mean equal to 3 minutes. Suppose the arrival rate is increased by 20%. Determine the change in the average number of customers in the system and the average time a customer spends in the system.
Consider the M/M/16 queuing system λ=8 μ=14 and p = λ/(sμ) (a) average number of customers in the system (b) average waiting time of each customer who enters the system (c) probability that all se...
Consider the M/M/16 queuing system λ=8 μ=14 and p = λ/(sμ) (a) average number of customers in the system (b) average waiting time of each customer who enters the system (c) probability that all servers are occupied We were unable to transcribe this imageWe were unable to transcribe this imagePU > s) = (s!)(1-p) We were unable to transcribe this image PU > s) = (s!)(1-p)
Problem 8: 10 points Consider a queuing system M/M/1 with one server. Customer arrivals form a...
Problem 8: 10 points Consider a queuing system M/M/1 with one server. Customer arrivals form a Poisson process with the intensity A 15 per hour. Service times are exponentially distributed with the expectation3 minutes Assume that the number of customers at t-0, has the stationary distribution. 1. Find the average queue length, (L) 2. What is the expected waiting time, (W), for a customer? 3. Determine the expected number of customers that have completed their services within the 8-hour shift
Consider a simple queuing system in which customers arrive randomly such that the time between successive...
Consider a simple queuing system in which customers arrive randomly such that the time between successive arrivals is exponentially distributed with a rate parameter l = 2.8 per minute. The service time, that is the time it takes to serve each customer is also Exponentially distributed with a rate parameter m = 3 per minute. Create a Matlab simulation to model the above queuing system by randomly sampling time between arrivals and service times from the Exponential Distribution. If a...
Multiple Server Waiting Line Model Regional Airlines Assumptions Poisson Arrivals Exponential Service Times Number of Servers...
Multiple Server Waiting Line Model Regional Airlines Assumptions Poisson Arrivals Exponential Service Times Number of Servers Arrival Rate Service Rate For Each Server Operating Characteristics 4 Probability that no customer are in the system, Po 5 Average number of customer in the waiting line, L 6 Average number of customer in the system, L 7 Average time a customer spends in the waiting line, W 18 Average time a customer spends in the system, W 19 Probability an arriving customer...
Question 7. (15 marks] Consider the discrete time system given by the state equation 07 x4...
Question 7. (15 marks] Consider the discrete time system given by the state equation 07 x4 + 11-18 8/11 - 10/n VIK) = 10 11 **) 1. [3 marks) Determine if the system is (a) Lyapunov state, syptereally ) Bounded input Bounded Output (BIBO) stable. Provide brief explanations 2. (8 marks) Design a discrete-time state feedback control law of the form - Kxkl by finding the gain K to place the closed-loop eigenvalues at 0.5 3. [4 marks) Suppose the...
damped forced mass-spring system with m 2, and k 26, under the 2 Consider a influence...
damped forced mass-spring system with m 2, and k 26, under the 2 Consider a influence of an external force F(t)= 82 cos (4t) 1, 7 = a) (8 points) Find the position u(t) of the mass at any time t, if u(0) 6 and u'(0) = 0. b) (4 points) Find the transient solution u(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time? We were unable to...
Problem 4 (25%) Consider the attitude control system of a rigid satellite shown in Figure 1.1....
Problem 4 (25%) Consider the attitude control system of a rigid satellite shown in Figure 1.1. Fig. 1.1 Satellite tracking control system In this problem we will only consider the control of the angle e (angle of elevation). The dynamic model of the rigid satellite, rotating about an axis perpendicular to the page, can be approximately written as: JÖ = tm - ty - bė where ) is the satellite's moment of inertia, b is the damping coefficient, tm is...
1. Consider a harmonic oscillator sitting in the ground state with a given spring constant ko m w...
1. Consider a harmonic oscillator sitting in the ground state with a given spring constant ko m were is constant). We want to change the system to raise the constant to 4ko. [N.B. You will have to use the equation versions of the eigenstates for this question since the system is changing a) Use the ideal instantaneous sudden approximation to find the probability that the system stays in the ground state. Does this approximation include selection rules? b) Assuming that...
The M/M/1 and M/M/1/K queuing system: Consider the M/M/1 and
M/M/1/K queuing systems [see in class notes]. For the M/M/1/K
system show that, for ρ < 1,
in class notes:
p" (1-p) п-0, 1,2, ..., К-1; р-— 1-р*а п, K+1 и N- Р_(К+1)pku К-+1 1-р*а 1-р M/M/1 Queuing System with Finite Capacity (M/M/1/K) Systems have a finite capacity for serving customers. The M/M/1 queuing system capable of supporting up to K customers is called an M/M/1/K queuing system. Arrivals at...
Problem 8: 10 points Consider a queuing system M/M/1 with one server. Customer arrivals form a Poisson process with the intensity A 15 per hour. Service times are exponentially distributed with the expectation3 minutes Assume that the number of customers at t-0, has the stationary distribution. 1. Find the average queue length, (L) 2. What is the expected waiting time, (W), for a customer? 3. Determine the expected number of customers that have completed their services within the 8-hour shift
Multiple Server Waiting Line Model Regional Airlines Assumptions Poisson Arrivals Exponential Service Times Number of Servers Arrival Rate Service Rate For Each Server Operating Characteristics 4 Probability that no customer are in the system, Po 5 Average number of customer in the waiting line, L 6 Average number of customer in the system, L 7 Average time a customer spends in the waiting line, W 18 Average time a customer spends in the system, W 19 Probability an arriving customer...
Question 7. (15 marks] Consider the discrete time system given by the state equation 07 x4 + 11-18 8/11 - 10/n VIK) = 10 11 **) 1. [3 marks) Determine if the system is (a) Lyapunov state, syptereally ) Bounded input Bounded Output (BIBO) stable. Provide brief explanations 2. (8 marks) Design a discrete-time state feedback control law of the form - Kxkl by finding the gain K to place the closed-loop eigenvalues at 0.5 3. [4 marks) Suppose the...
damped forced mass-spring system with m 2, and k 26, under the 2 Consider a influence of an external force F(t)= 82 cos (4t) 1, 7 = a) (8 points) Find the position u(t) of the mass at any time t, if u(0) 6 and u'(0) = 0. b) (4 points) Find the transient solution u(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time? We were unable to...
Problem 4 (25%) Consider the attitude control system of a rigid satellite shown in Figure 1.1. Fig. 1.1 Satellite tracking control system In this problem we will only consider the control of the angle e (angle of elevation). The dynamic model of the rigid satellite, rotating about an axis perpendicular to the page, can be approximately written as: JÖ = tm - ty - bė where ) is the satellite's moment of inertia, b is the damping coefficient, tm is...
1. Consider a harmonic oscillator sitting in the ground state with a given spring constant ko m were is constant). We want to change the system to raise the constant to 4ko. [N.B. You will have to use the equation versions of the eigenstates for this question since the system is changing a) Use the ideal instantaneous sudden approximation to find the probability that the system stays in the ground state. Does this approximation include selection rules? b) Assuming that...
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