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Answer :
An exponential service time with an average service rate of 22 customers per hour, the system is stable and has a well-defined steady state.
A single server queuing system with a Poisson arrival rate and exponential service time is commonly referred to as an M/M/1 queue. The "M" stands for the memoryless property of arrivals and service times, while the "1" represents the single server.
For this particular system, the average arrival rate is given as 14 customers per hour, which means that on average, 14 customers arrive in the system every hour. The average service rate is stated as 22 customers per hour, indicating that on average, the server can complete service for 22 customers within an hour.
To determine the stability of the system, we compare the arrival rate with the service rate. In this case, the arrival rate (14 customers per hour) is less than the service rate (22 customers per hour), indicating that the system is stable. If the arrival rate were greater than the service rate, the system would become unstable, resulting in an increasing number of customers in the queue over time.
In a stable M/M/1 queue, the steady state probabilities can be calculated, such as the probability of having a certain number of customers in the system or the average number of customers in the system. These calculations are based on queuing theory formulas, which take into account the arrival rate and service rate.
Overall, the given single server queuing system with a Poisson arrival rate of 14 customers per hour and an exponential service time with an average service rate of 22 customers per hour is stable and can be analyzed using queuing theory to determine various performance metrics.
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