In queuing theory, a discipline within the mathematical theory of probability, an M/D/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation.[1] Agner Krarup Erlang first published on this model in 1909, starting the subject of queuing theory.[2][3] An extension of this model with more than one server is the M/D/c queue
An M/D/1 queue is a stochastic process whose state space is the set {0, 1, 2, 3,} where the value corresponds to the number of customers in the system, including any currently in service.
Arrivals occur at rate λ according to a Poisson process and move the process …show more content…
By varying λ and µ, server utilization can assume any value between 0 and 1. Yet there is never any line. In general, variability of inter arrival and service times causes lines to fluctuate in length. Example: A physician who schedules patients every 10 minutes and spends Si minutes with the ith patient: Arrivals are deterministic, A1 = A 2 = … = λ-1 = 10. Services are stochastic, E (Si) = 9.3 min and V(S 0) = 0.81 min 2. On average, the physician's utilization = ρ = λ/µ = 0.93 < 1. Consider the system is simulated with service times: S1 = 9, S 2 = 12, S 3 = 9, S 4 = 9, S 5 = 9, The system becomes: The occurrence of a relatively long service time ( S 2 = 12) causes a waiting line to form …show more content…
Cars arrive at the gas station according to a Poisson process. The arrival rate is 20 cars per hour. Cars are served in order of arrival. The service time (i.e. the time needed for pumping and paying) is exponentially distributed. The mean service time is 2 minutes. (i) Determine the distribution, mean and variance of the number of cars at the gas station. (ii) Determine the distribution of the sojourn time and the waiting time. (iii) What is the fraction of cars that has to wait longer than 2 minutes? 40 An arriving car finding 2 cars at the station immediately leaves. (iv) Determine the distribution, mean and variance of the number of cars at the gas station. (v) Determine the mean sojourn time and the mean waiting time of all cars (including the ones that immediately leave the gas
An M/D/1 queue is a stochastic process whose state space is the set {0, 1, 2, 3,} where the value corresponds to the number of customers in the system, including any currently in service.
Arrivals occur at rate λ according to a Poisson process and move the process …show more content…
By varying λ and µ, server utilization can assume any value between 0 and 1. Yet there is never any line. In general, variability of inter arrival and service times causes lines to fluctuate in length. Example: A physician who schedules patients every 10 minutes and spends Si minutes with the ith patient: Arrivals are deterministic, A1 = A 2 = … = λ-1 = 10. Services are stochastic, E (Si) = 9.3 min and V(S 0) = 0.81 min 2. On average, the physician's utilization = ρ = λ/µ = 0.93 < 1. Consider the system is simulated with service times: S1 = 9, S 2 = 12, S 3 = 9, S 4 = 9, S 5 = 9, The system becomes: The occurrence of a relatively long service time ( S 2 = 12) causes a waiting line to form …show more content…
Cars arrive at the gas station according to a Poisson process. The arrival rate is 20 cars per hour. Cars are served in order of arrival. The service time (i.e. the time needed for pumping and paying) is exponentially distributed. The mean service time is 2 minutes. (i) Determine the distribution, mean and variance of the number of cars at the gas station. (ii) Determine the distribution of the sojourn time and the waiting time. (iii) What is the fraction of cars that has to wait longer than 2 minutes? 40 An arriving car finding 2 cars at the station immediately leaves. (iv) Determine the distribution, mean and variance of the number of cars at the gas station. (v) Determine the mean sojourn time and the mean waiting time of all cars (including the ones that immediately leave the gas