Chapter 20.6, Problem 2P

Explanation of Solution

Given:

• The total cost of employing a teller at a bank is $100 per day.
• A teller can serve an average of 60 customers per day.
• An average of 50 customers per day arrives at the bank.
• Both service times and inter arrival times are exponential.
• The delay cost per customer day is $100.

To determine

The number of tellers that the bank should hire is to be determined.

Calculating the number of tellers, the bank should hire:

The queuing system is a $\text{M|M|s|GD|}\infty \text{|}\infty$ system with exponential inter arrival times, exponential service times, s servers, general queue discipline, infinite capacity and infinite population size from which the customers are drawn.

Here,

Arrival rate, $\text{λ=50}$ customers/day

Service rate, $\text{μ=60}$ customers/day

Also, if there are s servers, then,

$\text{ρ = }\frac{\text{λ}}{\text{sμ}}\text{ = }\frac{\text{50}}{\text{60s}}\text{ = }\frac{\text{5}}{\text{6s}}$

In queuing optimization problem, the component of cost due to customers waiting in line is referred to as the delay cost. Thus, the bank wants to minimize,

$\frac{\text{Total}\text{Expected}\text{cost}}{\text{Day}}\text{ = }\frac{\text{Expected}\text{service}\text{cost}}{\text{Day}}\text{ + }\frac{\text{Expected}\text{delay}\text{cost}}{\text{Day}}\mathrm{...}\text{(1)}$

Since, the total cost of employing a tell at a bank is $100 per day, then,

$\frac{\text{Expected}\text{service}\text{cost}}{\text{Day}}\text{ = 100s}$

Case 1:

For number of servers,  $\text{s = 1}$.

$\frac{\text{Expected}\text{service}\text{cost}}{\text{Day}}\text{ = 100 }\times \text{ 1 = \$100}\text{.00}\mathrm{...}\text{(2)}$

The Hourly delay cost is defined as follows:

$\frac{\text{Expected}\text{delay}\text{cost}}{\text{Day}}\text{=}\left(\frac{\text{Expected}\text{delay}\text{cost}}{\text{Customer}}\right)\left(\frac{\text{Expected}\text{customers}}{\text{Day}}\right)$

Now, the delay cost per customer day is $\$100$. Thus,

$\frac{\text{Expected}\text{delay}\text{cost}}{\text{Customer}}\text{=}\left(\frac{\text{\$100}}{\text{Customer-day}}\right)\left(\text{average}\text{days}\text{customer}\text{spends}\text{in}\text{the}\text{system}\right)$

                                     $\text{= 100W}$

Also,

$\frac{\text{Expected}\text{customers}}{\text{Day}}\text{ = λ = 50}$

Hence,

$\frac{\text{Expected}\text{delay}\text{cost}}{\text{Day}}\text{=}\left(\text{\$100}\text{per}\text{customer - day}\right)\left(\text{50}\text{customers}\text{per}\text{day}\right)\text{W}\mathrm{...}\text{(3)}$

      $=100\times 50\times \text{W}$

Now, the average hours the customer spends in the system $\left(\text{W}\right)$ for a $\text{M|M|1}$ queuing system is as follows:

$\text{W=}\frac{\text{1}}{\text{μ-λ}}\text{=}\frac{\text{1}}{\text{60-50}}\text{=}\frac{\text{1}}{\text{10}}\text{days}$

Substituting $\text{W = 0}\text{.1}$ days in equation 3 to compute the expected delay cost per hour as follows:

  $\frac{\text{Expected}\text{delay}\text{cost}}{\text{Day}}\text{=}\left(\text{\$100}\text{per}\text{customer}\text{-}\text{day}\right)\left(\text{50}\text{customers}\text{per}\text{day}\right)\text{W}$

                   $\begin{array}{l}\text{=}\left(\text{\$100}\text{per}\text{customer}\text{-}\text{day}\right)\left(\text{50}\text{customers}\text{per}\text{day}\right)\text{(0}\text{.1}\text{day)}\\ \text{=\$500 }\mathrm{....}\text{(4)}\end{array}$

Now, substitute the value obtained from equation 2 and equation 4 in equation 1,

$\frac{\text{Total}\text{Expected}\text{cost}}{\text{Day}}\text{ = }\frac{\text{Expected}\text{service}\text{cost}}{\text{Day}}\text{ + }\frac{\text{Expected}\text{delay}\text{cost}}{\text{Day}}$

                            $=\$100.00+\$500.00=\$600.00$

Hence, the total cost per day for one server is $\$600$.

Case 2:

For number of servers, $\text{s = 2}$, the queuing system is $\text{M|M|2}$ queuing system is as follows:

$\text{ρ=50/120=0}\text{.41}$

$\frac{\text{Expected}\text{service}\text{cost}}{\text{Day}}=100s=100\times 2=\$200$