**Table 18.4: Infinite-Source Values for $ L_q $ and $ P_0 $ Given $ \lambda/\mu $ and $ M $**This table provides the infinite-source queueing values for $ L_q $ (average number of customers in line) and $ P_0 $ (probability of zero customers in the system) based on the ratio $ \lambda/\mu $ (arrival rate/service rate) and the number of servers $ M $.Each row of the table is organized as follows:- **$ \lambda/\mu $**: This column lists the ratio of arrival rate to service rate, ranging from 0.15 to 4.3.- **$ M $**: Indicates the number of servers in the system, ranging from 1 to 10.- **$ L_q $**: Represents the average number of customers in line for the queue.- **$ P_0 $**: Shows the probability that there are zero customers in the system.For each value of $ \lambda/\mu $, data is arranged under different values of $ M $, showing how $ L_q $ and $ P_0 $ vary with changes in these parameters. The table helps in understanding how queue lengths and idle probabilities fluctuate with different server counts and demand ratios, facilitating decision-making in service systems planning and optimization. **Problem 18-7 (Algo)**The manager of a regional warehouse must decide on the number of loading docks to request for a new facility in order to minimize the sum of dock costs and driver-truck costs. The manager has learned that each driver-truck combination represents a cost of $205 per day and that each dock plus loading crew represents a cost of $1,118 per day. Use Table 18.4.**a. How many docks should be requested if trucks arrive at the rate of four per day, each dock can handle five trucks per day, and both rates are Poisson?**- **Answer is complete and correct.** - Number of dock(s): **1****b. An employee has proposed adding new equipment that would speed up the loading rate to 5.71 trucks per day. The equipment would cost an additional $100 per day for each dock. What is the lowest daily total cost that can be achieved with the new equipment? (Round your cost amount to 2 decimal places and all other calculations to 3 decimal places.)**- **Answer is complete but not entirely correct.** - The daily total cost with the new equipment is: **$1,441.13***Explanations:*- Part (a) Analysis: The number of docks is calculated based on the arrival rate and handling capacity, ensuring the optimal number of docks to minimize costs.- Part (b) Analysis: An additional cost is considered with new equipment. The proposed loading rate adjustment and its financial impact on daily costs are evaluated. The given answer may need reassessment for full accuracy.

Given- Dock plus loading crew represents a cost = $1,118 per day Driver-truck combination cost = $205Part (A): Arrival rate λ = 4 trucks per day New service rate (μ)=5 trucks per day Utilization ρ=λμ=45=0.80 Refer the table and look up for λμ=0.80 Using Available 1, 2, 3 docks, i.e. M=1, value of Lq=3.2 M=2, value of Lq=1.52 M=3, value of Lq=0.19 For 1 Dock: Dock cost = $1118 L= Lq +λμ= 3.2 + 0.8 = 4 Waiting cost (Cw) = L×205 = 4×205 = $820 Total cost = Waiting cost + Dock cost = $1118+$820 = $1938 For 2 Docks: Dock cost = $1118×2 = $2236 L= Lq +λμ= 0.152 + 0.8 = 0.952 Waiting cost (Cw) = L×205 = 0.952×205 = $195.16 Total cost = Waiting cost + Dock cost = $2236+$195.16 = $2431.16 For 3 Docks: Dock cost = $1118×3 = $3354 L= Lq +λυ= 0.019 + 0.8 = 0.819 Waiting cost (Cw) = L×205 = 0.819×205 = $167.895 Total cost = Waiting cost + Dock cost = $3354+$167.895 = $3521.895 Since 1 dock is the economical system as cost is concerned. Therefore, 1 dock should be requested. Part (B): Using the new equipment- Arrival rate λ = 4 trucks per day New service rate (μ)=5.71 trucks per day Utilization ρ=λμ=45.71=0.701 Refer the table and look up for λμ=0.701 Using 1 docks, i.e. M=1, value of Lq=1.633So, L=Lq+λμ=1.633+0.701=2.334 Truck-driver combination cost (Cw) = $205 per day Dock plus loading crew cost (Cs) = $1118 per day Total cost per day = M×(Cs+Additional cost)+L×Cw = 1×(1118+100)+2.334×205 = $1696.47 Therefore, the daily total cost with the new equipment is $1696.47