Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alte Variables l Mean number of arrivals per time period (Poisson arrivals) m Mean number of people or items served per time period (exponential service times) k Number of people or items in the system of interest m Number of channels (i.e., service facilities) available in the system L W Lq Wq r P0 Pn>k Cs Cw t M/M/1 Queuing Equations Average number of people or items in the system (i.e., number in line plus number being served) Average time a person or item spends in the system (i.e., time spent in line plus time spent being ser Average number of people or items in the queue Average time person or item spends waiting in the queue Utilization factor (i.e. probability that the service facility is being used) Percent idle time (i.e., probability that no person or item is in the system) Probability that the number of people or items in the system is greater than k M/M/1 Costs Service cost for each channel (i.e., service facility) available in the system Cost of waiting Number of time periods per day Total service cost Total waiting cost (based upon average time in the system) Total waiting cost (based upon average time in the queue) Total cost (i.e, total service cost plus total waiting cost) - based upon average time in system Total cost (i.e, total service cost plus total waiting cost) - based upon average time in queue Total daily waiting cost Total daily service cost Total daily system cost (i.e. total daily waiting cost plus total daily service cost) Automobiles arrive at the drive-through window of the Elm Street branch of the local bank at the rate of 1 eve arrival rate and service times are exponentially distributed. s the calculated values. ed parameters. Do not alter the content of these cells. #1 What is the average number of customers in the system (in line plus 6.00 L 4 7.50 input data Question 1.00 2: What is the average time a customer spends in the system? 1.00 W 0.67 Question 3: What is the average number of customers in line \"behind\" the custo Wq 3.2 l/(m-l) 1/(m-l) l^2/(m(m-l)) l/(m(m-l)) l/m 1-(l/m) (l/m)^(k+1) 4.00 0.67 Question 5: What is the probability the customer service person is busy? 3.20 p 0.8 0.53 Question 0.80 6: What is the percent of time (or the probability) that there are zero customers at the serv 0.2000 Po 0.2 0.64 Question 7: What is the probability that there are more than two customers in th .8^2 0.64 m*Cs (l*W)*Cw (l*W q)*Cw (m*Cs) + ((l*W)*Cw) (m*Cs) + ((l*W q)*Cw) t*l*W q*Cw t*m*Cs (t*l*W q*Cw)+(t*m*Cs) $20.00 $27.00 input data 8.0000Question 8: What is the probability that there are exactly two customers in the system? $20.00 .8^2 * .2 0.128 $108.00 $86.40Question 9: What is the probability that there are two or less customers in the system? $128.00 0.36 $106.40 $691.20 Question 10: What is the probability that there are less than two custo $160.00 0.232 $851.20 cal bank at the rate of 1 every ten minutes. The average service time is 8 minutes. The Poisson distribution is appropriate for the in the system (in line plus being served)? s in line \"behind\" the customer receiving service (queue)? ice person is busy? zero customers at the serving window? e than two customers in the system? mers in the system? mers in the system? here are less than two customers in the system? on is appropriate for the Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alte Variables l m k m M/M/1 Queuing Equations L W Lq Wq r P0 Pn>k M/M/1 Costs Cs Cw t Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alter the con Variables Mean number of arrivals per time period (Poisson arrivals) Mean number of people or items served per time period (exponential service times) Number of people or items in the system of interest Number of channels (i.e., service facilities) available in the system M/M/1 Queuing Equations Average number of people or items in the system (i.e., number in line plus number being served) Average time a person or item spends in the system (i.e., time spent in line plus time spent being served) Average number of people or items in the queue Average time person or item spends waiting in the queue Utilization factor (i.e. probability that the service facility is being used) Percent idle time (i.e., probability that no person or item is in the system) Probability that the number of people or items in the system is greater than k M/M/1 Costs Service cost for each channel (i.e., service facility) available in the system Cost of waiting Number of time periods per day Total service cost Total waiting cost (based upon average time in the system) Total waiting cost (based upon average time in the queue) Total cost (i.e, total service cost plus total waiting cost) - based upon average time in system Total cost (i.e, total service cost plus total waiting cost) - based upon average time in queue Total daily waiting cost Total daily service cost Total daily system cost (i.e. total daily waiting cost plus total daily service cost) The salary and benefits for a bank teller at the Elm Street branch is $16.50 per hour. If it has been estimated that the w ulated values. eters. Do not alter the content of these cells. 6.00 7.50 1.00 1.00 Question 11: How many customers would enter the bank's drive-i 4 input data Question 12: How much total time would the customers spend wa 0.53000 Question 13: If only one lane is open, what is the total daily waiting time cost of the system? $1,664.00 l/(m-l) 4.00 * 1/(m-l) 0.67 l^2/(m(m-l)) 3.20 * l/(m(m-l)) 0.53 * l/m 0.80 Question 14: If only one lane is open what is the total wait and service cost of the system? 1-(l/m) 0.2000 $ 276.50 (l/m)^(k+1) 0.64 Question 15: How much total time would the customers spend in line waiting to be serviced (queue) during the entire da 2.12000 m*Cs (l*W)*Cw (l*W q)*Cw (m*Cs) + ((l*W)*Cw) (m*Cs) + ((l*W q)*Cw) t*l*W q*Cw t*m*Cs (t*l*W q*Cw)+(t*m*Cs) $16.50Question 16: If only one lane is open, what is the daily total cost of the queuing wait time? $65.00 input data 8.0000 $ 208.00 $16.50 $260.00 $208.00 $276.50 Question 17: If only one lane is open what are the total service and queuing $224.50 Is this based on average time in the system or queue? $1,664.00 $ 224.50 $132.00 $1,796.00 been estimated that the waiting time cost per hour is $65.00 per hour in line. Assume the bank is open 8 hours each day. ould enter the bank's drive-in system on a typical day? uld the customers spend waiting in line (system) during the entire day if one lane were open? f the system? ervice cost of the system? (queue) during the entire day if one lane were open? of the queuing wait time? he total service and queuing wait time costs? ystem or queue? 8 hours each day. Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not Variables l m m M/M/m Queuing Equations P0 L W Lq Wq r M/M/m Costs Cs Cw t Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alter th Variables Mean number of arrivals per time period (Poisson arrivals) Mean number of people or items served per time period (exponential service times) Number of channels available in system M/M/m Queuing Equations Probability there are zero people or items in the system Average number of people or items in the system (i.e., number in line plus number being served) Average time a person or item spends in the system (i.e., time spent in line plus time spent being served) Average number of people or items in the queue Average time person or item spends waiting in the queue Utilization factor (i.e. probability that the service facility is being used) Are these answers right? M/M/m Costs Service cost for each channel (i.e., service facility) available in the system Cost of waiting Number of time periods per day Total service cost Total waiting cost (based upon average time in the system) Total waiting cost (based upon average time in the queue) Total cost (i.e, total service cost plus total waiting cost) - based upon average time in system Total cost (i.e, total service cost plus total waiting cost) - based upon average time in queue Total daily waiting cost Total daily service cost Total daily system cost (i.e. total daily waiting cost plus total daily service cost) On Friday's the Elm Street branch opens a second lane (window) during its 8 hours of operation in order to accom the front of the line, it would go to the next available lane (window) for service. On Friday's customers arrive at the transaction every 3 minutes, following an exponential distribution. The salary and benefits for a bank teller at the E Question 22: What is the probability that there are no customers in line or being serviced? 0.6 Question 24: How much total time would customers spend waiting in line to be serviced (queue) on a typical day? 0 Question 26: What is the total daily cost of customers waiting in the system on a typical day? $17.33 Are these answers right? alculated values. How do I use this chart? meters. Do not alter the content of these cells. 10.00 20.00 2.00 Eq. 1 on p. 601 Eq. 2 on p. 601 L/l L-(l/m) Lq/l l/m 0.6000 0.53 0.05 0.03 0.00 0.50 input data Possible n values Actual n values 0 0 1 1 2 3 4 5 6 7 8 9 - Question 18: What is the average number of customers in the s 0.53 $16.50 Question 19: What is the average number of customers in line \"behind\" th $65.00 input data receiving service? 8.0000 0.03 m*Cs $33.00 (l*W)*Cw $34.67 Question 20: What is the average waiting time a customer spends in the s (l*W q)*Cw $2.17 0.05 (m*Cs) + ((l*W)*Cw) $67.67 (m*Cs) + ((l*W q)*Cw) $35.17 Question 21: What is the average time a customer is in the queue waiting t*l*W q*Cw $17.33 0 t*m*Cs $264.00 (t*l*W q*Cw)+(t*m*Cs) $281.33 ation in order to accommodate employees from a local manufacturing plant who want to deposit their paychecks. When a cust customers arrive at the rate of about 10 every hour according to a Poisson distribution and, on average, each customer servic or a bank teller at the Elm Street branch is $16.50 per hour. If it has been estimated that the waiting time cost per hour is $65.0 Question 23: What percentage of the time are the customer service persons busy? 0.5 n a typical day? Question 25: What is the total daily cost of customers waiting in line to be serviced (queue) on a ty $2.17 Question 27: What is the total daily cost of customers waiting in the system plus the cost of service $67.67 w do I use this chart? 1/n! 1.0000 1.0000 - (l/m)^n 1.0000 0 0.5000 0 1 1 1 1 1 1 1 1 SUBTOTAL (1/n!)*((l/m)^n) 1.0000 0.5000 1.5000 he average number of customers in the system? number of customers in line \"behind\" the customer(s) waiting time a customer spends in the system? time a customer is in the queue waiting to be serviced? deposit their paychecks. When a customer in the single line reaches nd, on average, each customer service person can process a he waiting time cost per hour is $65.00 per hour in line. ng in line to be serviced (queue) on a typical day? ng in the system plus the cost of service on a typical day? Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alte Variables l m k m M/M/1 Queuing Equations L W Lq Wq r P0 Pn>k M/M/1 Costs Cs Cw t Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alter the con Variables Mean number of arrivals per time period (Poisson arrivals) Mean number of people or items served per time period (exponential service times) Number of people or items in the system of interest Number of channels (i.e., service facilities) available in the system M/M/1 Queuing Equations Average number of people or items in the system (i.e., number in line plus number being served) Average time a person or item spends in the system (i.e., time spent in line plus time spent being served) Average number of people or items in the queue Average time person or item spends waiting in the queue Utilization factor (i.e. probability that the service facility is being used) Percent idle time (i.e., probability that no person or item is in the system) Probability that the number of people or items in the system is greater than k DO I USE THIS MODEL, MD1, OR SEVERAL OF EACH AND AVERAGE THEIR ANSWERS? M/M/1 Costs Service cost for each channel (i.e., service facility) available in the system Cost of waiting Number of time periods per day Total service cost Total waiting cost (based upon average time in the system) Total waiting cost (based upon average time in the queue) Total cost (i.e, total service cost plus total waiting cost) - based upon average time in system Total cost (i.e, total service cost plus total waiting cost) - based upon average time in queue Total daily waiting cost Total daily service cost Total daily system cost (i.e. total daily waiting cost plus total daily service cost) The Main Street branch of the bank is much larger than the Elm Street location and services more customers on a dai window. On average each customer service representative can service 6 customers every fifteen minutes at a cost of waiting time cost per hour is $70.00 per hour in line. The branch operates one lane for the entire 8 hours per day but o minutes and the arrival rate for peak time increases to 6 customers every 15 minutes. ulated values. eters. Do not alter the content of these cells. Question 28: What is the weighted average number of customers in the 10.00 20.00 1.00 2.00 input data Question 29: What is the weighted average number of customers in the Question 30: What is the weighted average time a customer spends in the system? l/(m-l) 1.00 * 1/(m-l) 0.10 Question 31: What is the weighted average time a customer spends in the queue? l^2/(m(m-l)) 0.50 * l/(m(m-l)) 0.05 * l/m Question 0.5032: What is the total (non-weighted) average wait time customers spend in the system on a ty 1-(l/m) 0.5000 (l/m)^(k+1) 0.25 Question 33: What is the total (non-weighted) average wait time customers spend in the queue on a typical da m*Cs (l*W)*Cw (l*W q)*Cw (m*Cs) + ((l*W)*Cw) (m*Cs) + ((l*W q)*Cw) t*l*W q*Cw t*m*Cs (t*l*W q*Cw)+(t*m*Cs) $22.50 Question 34: What is the total daily service cost? $70.00 input data 8.0000 Question $45.00 35: What is the total daily cost of customers waiting in the system on a typical day? $70.00 $35.00 $115.00 $80.00 Question 36: What is the total daily cost of customers waitin $280.00 system plus the cost of service on a typical day? $360.00 $640.00 s more customers on a daily base. Additionally it has upgraded it computer services and facilities making it quicker to service cus fteen minutes at a cost of salary and benefits for a bank teller is $22.50 per hour. The bank is open 8 hours each day. If it has been ntire 8 hours per day but opens a second lane during peak times four hours per day. Assume the arrival rate for off-peak hours is erage number of customers in the system? erage number of customers in the queue? spends in the queue? omers spend in the system on a typical day? spend in the queue on a typical day? e system on a typical day? otal daily cost of customers waiting in the making it quicker to service customers at its drive in 8 hours each day. If it has been estimated that the rrival rate for off-peak hours is 5 customers every 20 Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alte Variables l m k m M/M/1 Queuing Equations L W Lq Wq r P0 Pn>k M/M/1 Costs Cs Cw t Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alter the con Variables Mean number of arrivals per time period (Poisson arrivals) Mean number of people or items served per time period (exponential service times) Number of people or items in the system of interest Number of channels (i.e., service facilities) available in the system DO I USE THIS MODEL, MD1, OR SEVERAL OF EACH AND AVERAGE THEIR ANSWERS? M/M/1 Queuing Equations Average number of people or items in the system (i.e., number in line plus number being served) Average time a person or item spends in the system (i.e., time spent in line plus time spent being served) Average number of people or items in the queue Average time person or item spends waiting in the queue Utilization factor (i.e. probability that the service facility is being used) Percent idle time (i.e., probability that no person or item is in the system) Probability that the number of people or items in the system is greater than k M/M/1 Costs Service cost for each channel (i.e., service facility) available in the system Cost of waiting Number of time periods per day Total service cost Total waiting cost (based upon average time in the system) Total waiting cost (based upon average time in the queue) Total cost (i.e, total service cost plus total waiting cost) - based upon average time in system Total cost (i.e, total service cost plus total waiting cost) - based upon average time in queue Total daily waiting cost Total daily service cost Total daily system cost (i.e. total daily waiting cost plus total daily service cost) Your bank has decided to open a new drive-in widow complex at the Water Street location that is more accessible to it personnel that can service customers at the rate of 5 customers every twenty minutes. The plan is to keep one lane o open for 3 hours during afternoon/evening peak times (2:00 PM - 5:00 PM). Assume the arrival rates for off-peak times arrival rate for peak time increases to 6 customers every 20 minutes. Because this location is downtown and services estimates the wait time cost at $80.00 per hour. ulated values. eters. Do not alter the content of these cells. 10.00 20.00 1.00 2.00 Question 37: What is the weighted average number of customers in the system Question 38: What is the weighted average number of customers in the queu input data Question 39: What is the weighted average time a customer spends in the sys l/(m-l) 1/(m-l) l^2/(m(m-l)) l/(m(m-l)) l/m 1-(l/m) (l/m)^(k+1) 1.00 * Question 40: What is the weighted average time a customer spends in the que 0.10 0.50 * 0.05Question 41: #41What is the total (non-weighted average) wait time customers spend in the syste 0.50 0.5000 0.25Question 42: What is the total (non-weighted average) wait time customers spend in the queu m*Cs (l*W)*Cw (l*W q)*Cw (m*Cs) + ((l*W)*Cw) (m*Cs) + ((l*W q)*Cw) t*l*W q*Cw t*m*Cs (t*l*W q*Cw)+(t*m*Cs) Question 43: What is the total daily service cost? $22.50 $70.00 input data 8.0000 Question 44: What is the total daily cost of customers waiting in the system on a typ $45.00 $70.00 $35.00 Question 45: What is the total daily cost of customers waiting in the system plus the cost of ser $115.00 $80.00 $280.00 $360.00 $640.00 hat is more accessible to its downtown customers. The new branch will operate three lanes and will be manned using customer se plan is to keep one lane open all day (9:00 AM - 5:00 PM), a second lane open for 2 hours during lunch (Noon - 2:00 PM) and a thi val rates for off-peak times is 2 customers every 10 minutes, the arrival rate during lunch is 3.5 customers every 15 minutes, and t is downtown and services a more affluent client, the bank pays its customer service personnel $31.25 per hour (including benefits mber of customers in the system? umber of customers in the queue? e a customer spends in the system? e a customer spends in the queue? e customers spend in the system on a typical day? me customers spend in the queue on a typical day? daily service cost? s waiting in the system on a typical day? the system plus the cost of service on a typical day? be manned using customer service ch (Noon - 2:00 PM) and a third lane mers every 15 minutes, and the 5 per hour (including benefits) and Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not Variables l m m M/M/m Queuing Equations P0 L W Lq Wq r M/M/m Costs Cs Cw t Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alter th Variables Mean number of arrivals per time period (Poisson arrivals) Mean number of people or items served per time period (exponential service times) Number of channels available in system M/M/m Queuing Equations Probability there are zero people or items in the system Average number of people or items in the system (i.e., number in line plus number being served) Average time a person or item spends in the system (i.e., time spent in line plus time spent being served) Average number of people or items in the queue Average time person or item spends waiting in the queue Utilization factor (i.e. probability that the service facility is being used) M/M/m Costs Service cost for each channel (i.e., service facility) available in the system Cost of waiting Number of time periods per day Total service cost Total waiting cost (based upon average time in the system) Total waiting cost (based upon average time in the queue) Total cost (i.e, total service cost plus total waiting cost) - based upon average time in system Total cost (i.e, total service cost plus total waiting cost) - based upon average time in queue Total daily waiting cost Total daily service cost Total daily system cost (i.e. total daily waiting cost plus total daily service cost) On Friday's the Elm Street branch opens a second lane (window) during its 8 hours of operation in order to accommodate the front of the line, it would go to the next available lane (window) for service. On Friday's customers arrive at the rate of a transaction every 3 minutes, following an exponential distribution. The salary and benefits for a bank teller at the Elm Stree Question 22: What is the probability that there are no customers in line or being serviced? 0.6 Question 24: How much total time would customers spend waiting in line to be serviced (queue) on a typical day? 0 Question 26: What is the total daily cost of customers waiting in the system on a typical day? $17.33 alculated values. meters. Do not alter the content of these cells. 10.00 20.00 2.00 Eq. 1 on p. 601 Eq. 2 on p. 601 L/l L-(l/m) Lq/l l/m 0.6000 0.53 0.05 0.03 0.00 0.50 input data Possible n values Actual n values 0 0 1 1 2 3 4 5 6 7 8 9 - Question 18: What is the average number of customers in the s 0.53 $16.50 Question 19: What is the average number of customers in line \"behind\" th $65.00 input data receiving service? 8.0000 0.03 m*Cs $33.00 (l*W)*Cw $34.67 Question 20: What is the average waiting time a customer spends in the s (l*W q)*Cw $2.17 0.05 (m*Cs) + ((l*W)*Cw) $67.67 (m*Cs) + ((l*W q)*Cw) $35.17 Question 21: What is the average time a customer is in the queue waiting t*l*W q*Cw $17.33 0 t*m*Cs $264.00 (t*l*W q*Cw)+(t*m*Cs) $281.33 order to accommodate employees from a local manufacturing plant who want to deposit their paychecks. When a customer in the sing ers arrive at the rate of about 10 every hour according to a Poisson distribution and, on average, each customer service person can pro nk teller at the Elm Street branch is $16.50 per hour. If it has been estimated that the waiting time cost per hour is $65.00 per hour in lin Question 23: What percentage of the time are the customer service persons busy? 0.5 n a typical day? Question 25: What is the total daily cost of customers waiting in line to be serviced (queue) on a ty $2.17 Question 27: What is the total daily cost of customers waiting in the system plus the cost of service $67.67 (l/m)^n 1.0000 0 0.5000 0 1 1 1 1 1 1 1 1 SUBTOTAL he average number of customers in the system? 1/n! 1.0000 1.0000 - (1/n!)*((l/m)^n) 1.0000 0.5000 1.5000 number of customers in line \"behind\" the customer(s) waiting time a customer spends in the system? time a customer is in the queue waiting to be serviced? aychecks. When a customer in the single line reaches , each customer service person can process a me cost per hour is $65.00 per hour in line. ng in line to be serviced (queue) on a typical day? ng in the system plus the cost of service on a typical day? Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not Variables l m m M/D/1 Queuing Equations L W Lq Wq r P0 M/D/1 Costs Cs Cw t Change the value in yellow highlighted cells to observe how changing the parameter affects the calculated values. Blue highlighted cells contain formulas that calcuate certain values based upon the provided parameters. Do not alter the Variables Mean number of arrivals per time period (Poisson arrivals) Mean number of people or items served per time period (constant service time) Number of channels (i.e., service facilities) available in the system M/D/1 Queuing Equations Average number of people or items in the system (i.e., number in line plus number being served) Average time a person or item spends in the system (i.e., time spent in line plus time spent being serv Average number of people or items in the queue Average time person or item spends waiting in the queue Average server utilization Probability system is empty (i.e., percentage of time system is empty) M/D/1 Costs Service cost for each channel (i.e., service facility) available in the system Cost of waiting Number of time periods per day Total service cost Total waiting cost (based upon average time in the system) Total waiting cost (based upon average time in the queue) Total cost (i.e, total service cost plus total waiting cost) - based upon average time in system Total cost (i.e, total service cost plus total waiting cost) - based upon average time in queue Total daily waiting cost Total daily service cost Total daily system cost (i.e. total daily waiting cost plus total daily service cost) the calculated values. d parameters. Do not alter the content of these cells. 6.00 7.50 1.50 Lq + l/m W q + 1/m l^2/(2m*(m-l)) l/((1*m)*(m-l)) l/m 1-r 2.40 0.40 1.60 0.27 0.80 0.20 m*Cs (l*W)*Cw (l*W q)*Cw (m*Cs) + ((l*W)*Cw) (m*Cs) + ((l*W q)*Cw) t*l*W q*Cw t*m*Cs (t*l*W q*Cw)+(t*m*Cs) $50.00 $60.00 24.0000 $75.00 $144.00 $96.00 $219.00 $171.00 $2,304.00 $1,800.00 $4,104.00 input data input data