Question

In the \(\mathrm{M} / \mathrm{M} / \mathrm{s}\) queue if you allow the service rate to depend on the number in the system (but in such a way so that it is ergodic), what can you say about the output process? What can you say when the service rate \(\mu\) remains unchanged but \(\lambda>s \mu ?\)

Short answer

In an M/M/s queue with a service rate that depends on the number of customers in the system in such a way that it remains ergodic, the output process will still have the Markov property and memoryless property. However, if the arrival rate 𝜆 is greater than s𝜇 with 𝜇 remaining unchanged, the queue will become unstable, leading to indefinitely increasing waiting times and an unpredictable output process.

Step-by-step solution

Understand the M/M/s queue model

An M/M/s queue is a queueing model where arrivals follow a Poisson process (Markovian inter-arrival time), service times follow an exponential distribution (Markovian service time), and there are "s" servers in the system.

Analyze the service rate depending on the number in the system

Assuming the service rate depends on the number of customers in the system in such a way that the process remains ergodic, the output process will still have the Markov property. Thus, while the service time distribution may vary with the number of customers, the output process will continue to have memoryless property.

Analyze the situation when 𝜆 > s𝜇

In this case, the arrival rate (𝜆) is greater than the maximum possible service rate (s𝜇) for all customers in the system. Therefore, the queue will become unstable, and the average number of customers in the system and the waiting time in the system will indefinitely increase. In this case, the output process is unpredictable, as customers may not ever leave the system. In conclusion, when the service rate depends on the number of customers in the system, the output process will still follow the Markov property and will be ergodic if the conditions are maintained. However, when the arrival rate is higher than the product of the number of servers s and the service rate 𝜇 while keeping 𝜇 unchanged, the queue will become unstable, and it's impossible to predict the output process.