$\begin{array}{l}
\begin{array}{|l|l|}
\hline \text { Priority }&\text { Average Arrival Rate (exponential interarrival times) }\
\hline \text { High } & 3 \text { per hour } \
\hline \text { Low } & 5 \text { per hour } \
\hline
\end{array}
\end{array}$
$\begin{array}{|l|l|}
\hline Nurrber~ of~ servers:&5\
\hline Service~ rate:&2~ per~ hour ~(exponential ~service~ times)\
\hline
\end{array}$
What is the average number of high priority items waiting in line for service?
A) 0
B) 0.24
C) 1.74
D) 1.98
E) 2.22

Question

$\begin{array}{l}
\begin{array}{|l|l|}
\hline \text { Priority }&\text { Average Arrival Rate (exponential interarrival times) }\
\hline \text { High } & 3 \text { per hour } \
\hline \text { Low } & 5 \text { per hour } \
\hline
\end{array}
\end{array}$
$\begin{array}{|l|l|}
\hline Nurrber~ of~ servers:&5\
\hline Service~ rate:&2~ per~ hour ~(exponential ~service~ times)\
\hline 
\end{array}$
What is the average number of high priority items waiting in line for service?
A) 0
B) 0.24
C) 1.74
D) 1.98
E) 2.22

Quizplus · Accepted Answer

The average number of high priority items waiting in line can be calculated using the formula for the average number of customers in the queue (Lq) in an M/M/c queue, where arrivals follow a Poisson process, service times are exponentially distributed, and there are c servers. Given the priority for high priority items, we can treat the system as prioritizing their service over low priority items. The total arrival rate (λ) is 3 + 5 = 8 per hour, and the service rate (μ) is 2 per hour per server with 5 servers, so the combined service rate is 10 per hour. However, since we're interested in the high priority items specifically and they have a separate arrival rate, we use their arrival rate (λ_high = 3) for calculations relevant to them. The utilization factor (ρ) is λ / (cμ) for the total system, but for calculating the queue for high priority items specifically, we adjust our approach to account for their prioritization, effectively treating them as if they are served in a system with reduced capacity due to the presence of low priority items as well. The exact calculation of Lq for priority queues can be complex and depends on the specifics of how the prioritization is implemented, but a simplified approach to estimate the impact on high priority items might involve calculating the effective load they present and the reduced capacity available to them due to the service of low priority items. Without the exact formula for a priority queue being specified, and given the information provided, a detailed calculation involving queueing theory principles specific to priority queues would be needed to accurately determine the average number of high priority items waiting. However, the choice provided here suggests an approximation or a specific calculation relevant to the given system's parameters, leading to the selection of an answer that would typically require applying queueing theory formulas that account for the priority mechanism's impact on service dynamics.

Nurrber of servers:5Service rate:2 per hour (exponential service times)\begin{array}{|l|l|}\hline Nurrber~ of~ servers:&5\\\hline Service~ rate:&2~ per~ hour ~(exponential ~service~ times)\\\hline \end{array}Nurrber of servers:Service rate:​52 per hour (exponential service times)​​

$\begin{array}{|l|l|}\hline Nurrber~ of~ servers:&5\\\hline Service~ rate:&2~ per~ hour ~(exponential ~service~ times)\\\hline \end{array}$