this the same question. note please this question is not the same as the one in the text book.

Design a Toll Plaza System and its Work Schedule Western Motorway Authority (WMA) operates a network of toll ronds connecting three urba ized areas. Each of its toll collection plazas offers a choice of payment methods to approaching drivers, including WMA FAST, a subscription service that allows drivers to pay wirelessly as they pass through a special toll lane without stopping. Exist 53 Toll Plaza is a typical WMA facility, equipped with six toll lanes, including two dedicated to WMA FAST subscribers only. The remaining five lanes are equipped with booths from which WMA attendants collect tolls manually, known as cash lanes even though attendants can process credit and debit card payments for an additional fee. Since exit 53 in the end point of a long commuter route into a city center, toll prices are too high at the plan to make automated coin collection a practical choice, although WMA does use this technology elsewhere. While exit 53 is equipped with six toll lanes, WMA only operates all six lanes during special events. The two fully automated WMA FAST lanes are always open to arriving subscribers and can process an average of one vehicle every 4 seconds. Attendants in the cash lanes generally average one vehicle every 20 seconds when busy, and WMA usually varies the number of open cash lanes depending on traffic conditions. During the 24 hours of a typical weekday, exit 53 cycles between three levels of traffic conditions, described in the Weekly Data for Exit 53 table. Overnight midnight to 5 a.m. 8 p.m. to midnight Table 1: Weekday Data for Exit 53 Penk Hours of penk Hours Relevant time periods 6am to 9 m m . to 6 m. p.m. to 7 p.m. 9 a.m. to 4 p.m. 7 pm to 8 p.m. Proportion of WMA FAST subscribers Average number of cars 2.000 600 per hour 802 70% 60% Approaching drivers who subscribe to WMA FAST follow brightly colored markers toward their dedicated lane, while the remaining drivers sort themselves evenly into short qurues before the open cash lanes. Since frequent users of the toll plaza are more likely to subscribe to WMA FAST the proportion of subscribers is highest during commute times and lowest overnight. One serious concern for WMA is tallback, or the length of the queue of vehicles that forms in front of a busy cash lane tollbooth. If WMA has too few cash lanes open and the resulting tailback grow long, this creates unsafe congestion in the approach are where drivers slow their vehicles and choone lane. To maintain a safe approach and, WMA ha determine that this fluctuating tailback should not average more than four waiting vehicles per lane. To open a cash lane, WMA must schedule an attendant, who will work a 4-hour shift in that lane. On weekdays, attendant can be schedule to start work at midnight, 4a.m., 8 a.m., noon, 4 p.m., or 8 p.m.. Design jointly the capacity for cash lanes and a work schedule for attendants by considering the scheduling restrictions and the safety requirement in the approach area. Answer the following questions: 1. Perform an analysis of cash lane using the M/M/1 model, determine what is the minimum number of cash lanes that need to open during each of the three traffic conditions on a weekday, to keep average tailback down to four vehicles or less? 2. How many attendants should start work at each of the 4-hour intervals described to ensure the operation of the minimize number of cash lanes? 3. Consider the probability that an arriving vehicle will find an idle cash lane open at exit 53, and can drive up and pay without waiting. Using your recommended weekday staffing schedule, when is the probability at its highest? When is it at its lowest? 4. Some people in WMA Management are concerned over the high cost of WMA FAST lane installations. What is the overall utilization of the WMA FAST lanes at exit 53? Design a Toll Plaza System and its Work Schedule Western Motorway Authority (WMA) operates a network of toll ronds connecting three urba ized areas. Each of its toll collection plazas offers a choice of payment methods to approaching drivers, including WMA FAST, a subscription service that allows drivers to pay wirelessly as they pass through a special toll lane without stopping. Exist 53 Toll Plaza is a typical WMA facility, equipped with six toll lanes, including two dedicated to WMA FAST subscribers only. The remaining five lanes are equipped with booths from which WMA attendants collect tolls manually, known as cash lanes even though attendants can process credit and debit card payments for an additional fee. Since exit 53 in the end point of a long commuter route into a city center, toll prices are too high at the plan to make automated coin collection a practical choice, although WMA does use this technology elsewhere. While exit 53 is equipped with six toll lanes, WMA only operates all six lanes during special events. The two fully automated WMA FAST lanes are always open to arriving subscribers and can process an average of one vehicle every 4 seconds. Attendants in the cash lanes generally average one vehicle every 20 seconds when busy, and WMA usually varies the number of open cash lanes depending on traffic conditions. During the 24 hours of a typical weekday, exit 53 cycles between three levels of traffic conditions, described in the Weekly Data for Exit 53 table. Overnight midnight to 5 a.m. 8 p.m. to midnight Table 1: Weekday Data for Exit 53 Penk Hours of penk Hours Relevant time periods 6am to 9 m m . to 6 m. p.m. to 7 p.m. 9 a.m. to 4 p.m. 7 pm to 8 p.m. Proportion of WMA FAST subscribers Average number of cars 2.000 600 per hour 802 70% 60% Approaching drivers who subscribe to WMA FAST follow brightly colored markers toward their dedicated lane, while the remaining drivers sort themselves evenly into short qurues before the open cash lanes. Since frequent users of the toll plaza are more likely to subscribe to WMA FAST the proportion of subscribers is highest during commute times and lowest overnight. One serious concern for WMA is tallback, or the length of the queue of vehicles that forms in front of a busy cash lane tollbooth. If WMA has too few cash lanes open and the resulting tailback grow long, this creates unsafe congestion in the approach are where drivers slow their vehicles and choone lane. To maintain a safe approach and, WMA ha determine that this fluctuating tailback should not average more than four waiting vehicles per lane. To open a cash lane, WMA must schedule an attendant, who will work a 4-hour shift in that lane. On weekdays, attendant can be schedule to start work at midnight, 4a.m., 8 a.m., noon, 4 p.m., or 8 p.m.. Design jointly the capacity for cash lanes and a work schedule for attendants by considering the scheduling restrictions and the safety requirement in the approach area. Answer the following questions: 1. Perform an analysis of cash lane using the M/M/1 model, determine what is the minimum number of cash lanes that need to open during each of the three traffic conditions on a weekday, to keep average tailback down to four vehicles or less? 2. How many attendants should start work at each of the 4-hour intervals described to ensure the operation of the minimize number of cash lanes? 3. Consider the probability that an arriving vehicle will find an idle cash lane open at exit 53, and can drive up and pay without waiting. Using your recommended weekday staffing schedule, when is the probability at its highest? When is it at its lowest? 4. Some people in WMA Management are concerned over the high cost of WMA FAST lane installations. What is the overall utilization of the WMA FAST lanes at exit 53