To determine the average time a customer experiences from the moment they place a delivery order until the groceries arrive at their location, we need to account for all time components involved:

Average Time Components:

Driver's Arrival Time at the Grocery Store: $10$ minutes

Shopping Time: $20$ minutes

Delivery Time from Grocery Store to Customer: $30$ minutes

Thus, the average total time spent in the service process, not accounting for queueing, is:

Total$$Time

$=$

$10$

$$minutes

$+$

$20$

$$minutes

$+$

$30$

$$minutes

$=$

$60$

$$minutes

Total$$Time$=10$minutes$+20$minutes$+30$minutes$=60$minutes

Service Rate Calculation:

The average rate of delivery orders is $3\mathrm{.}5$ orders per hour.

Therefore, the average time between orders $($inter$-$arrival time$)$ is:

Time$$Between$$Orders

$=$

$60$

$$minutes

$3\mathrm{.}5$

$$orders$/$hour

$$

$17\mathrm{.}14$

$$minutes$/$order

Time$$Between$$Orders$=$

$3\mathrm{.}5$orders$/$hour

$60$minutes

$$

$17\mathrm{.}14$minutes$/$order

Queueing Time Analysis:

Given:

The number of delivery vans $($servers$)$

$$

$=$

$7$

S$=7$

The average service time $($total time for a single order$)$

$$

$$

$=$

$60$

$$minutes

T

s

$$

$=60$minutes

The average arrival rate

$$

$=$

$3\mathrm{.}5$

$$orders$/$hour

$\backslash$lambda $=3\mathrm{.}5$orders$/$hour

The average service rate

$$

$=$

$1$

$$

$$

$=$

$1$

$60$

$$orders$/$minute

$=$

$1$

$$order$/$hour

$\backslash$mu $=$

T

s

$$

$1$

$$

$=$

$60$

$1$

$$

$$orders$/$minute$=1$order$/$hour

We use the Multi$-$server Queueing System $($M$/$M$/$S$)$ model to estimate the average waiting time in the queue. The traffic intensity

$$

$\backslash$rho is:

$$

$=$

$$

$$

$$

$=$

$3\mathrm{.}5$

$7$

$\backslash$times

$1$

$60$

$=$

$3\mathrm{.}5$

$\backslash$times

$60$

$7$

$=$

$30$

$\backslash$rho $=$

S$\backslash$mu

$\backslash$lambda

$$

$=$

$7\backslash$times

$60$

$1$

$$

$3\mathrm{.}5$

$$

$=$

$7$

$3\mathrm{.}5\backslash$times $60$

$$

$=30$

Since

$$

$\backslash$rho is much less than $1,$ indicating that the system is underutilized, the waiting time in the queue is negligible.

Overall Time:

Considering the system is underutilized, the queueing time is minimal. Hence, the average time a customer experiences, from order placement to delivery, is approximately equal to the total service time, which is:

Average$$Time$$Experienced

$$

$60$

$$minutes

Average$$Time$$Experienced$60$minutes

Conclusion:

The average time a customer experiences from the moment they place a delivery order with Maria until the groceries arrive at their location is approximately $60$ minutes. This calculation assumes that the waiting time in the queue is minimal due to the underutilization of the delivery fleet.