Háskóli Íslands Véla-og iðnaðarverkfræði Hermun Verkefni 1. Sigríður Sigurðardóttir

Transcription

1 Háskóli Íslands Véla-og iðnaðarverkfræði Hermun 2006 Verkefni 1 Sigríður Sigurðardóttir sigridua@hi.is

2 1. liður Hermun endurtekin 1000 sinnum. Súlurit 180 Meðalbiðtími Fjöldi kúnna Tími [mín.] 2

3 180 Meðalfjöldi í röð Fjöldi kúnna Meðalfjöldi í röð 60 Nýting þjóns Fjöldi kúnna Nýting þjóns[%] 3

4 50 Heildarhermunartimi Fjöldi kúnna Tími[mín.] Quantile-quantile rit Meðalbiðtími QQ Plot of Sample Data versus Standard Normal 50 Meðalfjöldi í röð QQ Plot of Sample Data versus Standard Normal 60 Quantiles of Input Sample Quantiles of Input Sample Standard Normal Quantiles Standard Normal Quantiles 4

5 Nýting þjóns QQ Plot of Sample Data versus Standard Normal 1.1 Heildarhermunartími QQ Plot of Sample Data versus Standard Normal 1150 Quantiles of Input Sample Quantiles of Input Sample Standard Normal Quantiles Standard Normal Quantiles Quantile-quantile ritin sýna að nýting þjóns og heildarhermunartími fylgja normaldreifingu en meðalbiðtími og nýting þjóns ekki. Að endingu: Meðalbiðtími Meðalfjöldi í röð Nýting Heildarhermunartími þjóns Meðalgildi Miðgildi Staðalfrávik Stærsta gildi Minnsta gildi

6 2. liður Í öðrum lið er öðrum þjóni bætt við. Þá tekur þjónn nr. tvö við viðskiptavinum sem koma þegar þjónn nr. eitt er upptekinn. Hins vegar er viðskiptavinum alltaf vísað á nr. eitt ef báðir þjónar eru lausir. Súlurit 70 Meðalbiðtími Fjöldi kúnna Tími [mín.] 6

7 80 Nýting þjóns Fjöldi kúnna Nýting þjóns[%] 60 Nýting þjóns Fjöldi kúnna Nýting þjóns[%] 7

8 70 Meðalfjöldi í röð Fjöldi kúnna Meðalfjöldi í röð 60 Heildarhermunartimi Fjöldi kúnna Tími[mín.] 8

9 Quantile-quantile rit Meðalbiðtími Meðalfjöldi í röð Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal Standard Normal Quantiles Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal Standard Normal Quantiles Nýting þjóns 1 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal Standard Normal Quantiles Heildarhermunartími Nýting þjóns 2 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal Standard Normal Quantiles Standard Normal Quantiles Quantiles of Input Sample QQ Plot of Sample Data versus Standard Norm 9

10 Líkt og í fyrsta lið fylgja nýting þjóns og heildarhermunartími normaldreifingu en meðalbiðtími og meðalfjöldi í röð ekki. Meðalbiðtími Meðalfjöldi í Nýting Nýting Heildarhermunartí röð þjóns 1 þjóns 1 mi Meðalgildi Miðgildi Staðalfrávik Stærsta gildi Minnsta gildi Eins og sést á nýtingu þjónanna er mun meira álag á þjóni 1 enda er meira vísað á hann í þessu hermunarforriti. 3. liður Hér skal jafna álagið á þjónana. Kóði: /* External definitions for single-server queueing system. GNU GCC compile ( gcc -o mm1.exe mm1tpr.c -lm GNU C standard format style ( indent --honour-newlines -nbbo --line-length120 foo.c -o bar.c */ #include <stdio.h> #include <stdlib.h> #include <math.h> #define Q_LIMIT 100 /* Limit on queue length. */ #define BUSY 1 /* Mnemonics for server's being busy */ 10

11 #define IDLE 0 /* and idle. */ unsigned int SEED; int next_event_type, num_custs_delayed, num_delays_required, num_events, num_in_q, server_status[3]; double area_num_in_q, area_server_status[3], mean_interarrival, mean_service, sim_time, time_arrival[q_limit + 1], time_last_event, time_next_event[4], total_of_delays, i; double urand () /* Uniform variate generation function (0 1]. */ return ((1.0 + (double) rand ()) * (1.0 / ((double) RAND_MAX + 1.0))); double expon (double mean) function. */ /* Exponential variate generation /* Return an exponential random variate with mean "mean". */ return -mean * log (urand ()); initialize () /* Initialization function. */ /* Initialize the simulation clock. */ sim_time = 0.0; /* Initialize the state variables. */ server_status[1] = IDLE; server_status[2] = IDLE; num_in_q = 0; time_last_event = 0.0; /* Initialize the statistical counters. */ num_custs_delayed = 0; total_of_delays = 0.0; area_num_in_q = 0.0; 11

12 area_server_status[1] = 0.0; area_server_status[2] = 0.0; /* Initialize event list. Since no customers are present, the departure (service completion) event is eliminated from consideration. */ time_next_event[1] = sim_time + expon (mean_interarrival); time_next_event[2] = 1.0e+30; time_next_event[3] = 1.0e+30; timing () /* Timing function. */ int i; double min_time_next_event = 1.0e+29; next_event_type = 0; /* Determine the event type of the next event to occur. */ for (i = 1; i <= num_events; ++i) if (time_next_event[i] < min_time_next_event) min_time_next_event = time_next_event[i]; next_event_type = i; /* Check to see whether the event list is empty. */ if (next_event_type == 0) /* The event list is empty, so stop the simulation. */ printf ("\nevent list empty at time %f", sim_time); exit (1); /* The event list is not empty, so advance the simulation clock. */ sim_time = min_time_next_event; 12

13 arrive () /* Arrival event function. */ double delay; int server_id; /* Schedule next arrival. */ time_next_event[1] = sim_time + expon (mean_interarrival); /* Check to see whether server is busy. */ if (server_status[1]==busy && server_status[2]==busy) /* Server is busy, so increment number of customers in queue. */ ++num_in_q; /* Check to see whether an overflow condition exists. */ if (num_in_q > Q_LIMIT) /* The queue has overflowed, so stop the simulation. */ printf ("\noverflow of the array time_arrival at"); printf (" time %f", sim_time); exit (2); /* There is still room in the queue, so store the time of arrival of the arriving customer at the (new) end of time_arrival. */ time_arrival[num_in_q] = sim_time; else if ((server_status[1]==busy && server_status[2]==idle)) server_id = 2; else if (server_status[1]==idle && server_status[2]==busy) server_id = 1; else 13

14 server_id = rand ()%2 +1 ; /* Server is idle, so arriving customer has a delay of zero. (The following two statements are for program clarity and do not affect the results of the simulation.) */ delay = 0.0; total_of_delays += delay; /* Increment the number of customers delayed, and make server busy. */ ++num_custs_delayed; server_status[server_id] = BUSY; /* Schedule a departure (service completion). */ time_next_event[server_id + 1] = sim_time + expon (mean_service); depart (int server_id) function. */ /* Departure event int i; double delay; /* Check to see whether the queue is empty. */ if (num_in_q == 0) /* The queue is empty so make the server idle and eliminate the departure (service completion) event from consideration. */ server_status[server_id] = IDLE; time_next_event[server_id+1] = 1.0e+30; else /* The queue is nonempty, so decrement the number of customers in queue. */ --num_in_q; /* Compute the delay of the customer who is beginning service and update 14

15 the total delay accumulator. */ delay = sim_time - time_arrival[1]; total_of_delays += delay; /* Increment the number of customers delayed, and schedule departure. */ ++num_custs_delayed; time_next_event[server_id+1] = sim_time + expon (mean_service); /* Move each customer in queue (if any) up one place. */ for (i = 1; i <= num_in_q; ++i) time_arrival[i] = time_arrival[i + 1]; report () /* Report generator function. */ /* Compute and write estimates of desired measures of performance. */ printf ("%f\t", total_of_delays / num_custs_delayed); printf ("%f\t", area_num_in_q / sim_time); printf ("%f\t", area_server_status[1] / sim_time); printf ("%f\t", area_server_status[2] / sim_time); printf ("%f\n", sim_time); update_time_avg_stats () /* Update area accumulators for time-average statistics. */ double time_since_last_event; /* Compute time since last event, and update last-event-time marker. */ time_since_last_event = sim_time - time_last_event; time_last_event = sim_time; /* Update area under number-in-queue function. */ area_num_in_q += num_in_q * time_since_last_event; /* Update area under server-busy indicator function. */ 15

16 area_server_status[1] += server_status[1] * time_since_last_event; area_server_status[2] += server_status[2] * time_since_last_event; int main (int argc, char *argv[]) /* Main function. */ srand (time(null)); for(i=1;i<1000;i++) /* Read input parameters from command line arguments. */ if (5 == argc) /* convert input arguments to the double or integer */ mean_interarrival = atof (argv[1]); mean_service = atof (argv[2]); num_delays_required = atoi (argv[3]); SEED = atoi (argv[4]); else /* display some help to the user and exit */ /* printf ("usage: %s mean_interarrival mean_service num_delays_required SEED\n", argv[0]); printf ("example: %s \n", argv[0]);*/ return 1; /* Specify the number of events for the timing function. */ num_events = 3; /* Write report heading and input parameters. */ /*printf ("Single-server queueing system\n\n");*/ /*printf ("Random number seed %u\n\n", SEED);*/ /*printf ("Mean interarrival time%11.3f minutes\n\n", mean_interarrival);*/ 16

17 /*printf ("Mean service time%16.3f minutes\n\n", mean_service);*/ /*printf ("Number of customers%14d\n\n", num_delays_required);*/ /* Initialize the random number generator */ /* Initialize the simulation. */ initialize (); /* Run the simulation while more delays are still needed. */ while (num_custs_delayed < num_delays_required) /* Determine the next event. */ timing (); /* Update time-average statistical accumulators. */ update_time_avg_stats (); /* Invoke the appropriate event function. */ switch (next_event_type) case 1: arrive (); break; case 2: depart(1); break; case 3: depart(2); break; /* Invoke the report generator and end the simulation. */ report (); return 0; Byrjaði á að keyra forritið með 0.9 mín. meðalþjónustutíma Nú er jafnari nýting á þjónum 1 og 2 og álagið því nokkuð jafnt á þjónana: Meðaltal af nýtingu á þjóni 1 = Meðaltal af nýtingu á þjóni 2= Breytum meðalþjónustutíma í 0.5 mín. Fæ þá meira álag á þjón 2: Meðaltal af nýtingu á þjóni 1 = Meðaltal af nýtingu á þjóni 2=

18 4. liður 1.8 Til að sanna þetta þarf að skoða U á jöðrum bilsins, eða í 0 og 1. Ef U = 0 þá er β In U = β In 0 = Ef U = 1 þá er β In U = β In 1 = 0 En ef fallið β ln(1-u) er skoðað fæst: Er U = 0 þá er β ln(1-u) = β ln(1-0) = β ln1 = 0 Ef U = 1 þá er β ln(1-u) = β ln(1-1) = β ln0 = Því munar engu hvort við erum með β lnu eða β ln(1-u) þar eð við förum annars vegar frá óendanlegu í núll eða öfugt Kóði: /* External definitions for single-server queueing system. GNU GCC compile ( gcc -o mm1.exe mm1tpr.c -lm GNU C standard format style ( indent --honour-newlines -nbbo --line-length120 foo.c -o bar.c */ #include <stdio.h> #include <stdlib.h> #include <math.h> #define Q_LIMIT 100 /* Limit on queue length. */ #define BUSY 1 /* Mnemonics for server's being busy */ #define IDLE 0 /* and idle. */ unsigned int SEED; int next_event_type, num_custs_delayed, num_delays_required, num_events, num_in_q, server_status; double area_num_in_q, area_server_status, mean_interarrival, mean_service, sim_time, time_arrival[q_limit + 1], time_last_event, time_next_event[3], total_of_delays; double urand () /* Uniform variate generation function (0 1]. */ 18

19 return ((1.0 + (double) rand ()) * (1.0 / ((double) RAND_MAX + 1.0))); double expon (double mean) /* Exponential variate generation function. */ /* Return an exponential random variate with mean "mean". */ return -mean * log (urand ()); initialize () /* Initialization function. */ /* Initialize the simulation clock. */ sim_time = 0.0; /* Initialize the state variables. */ server_status = IDLE; num_in_q = 0; time_last_event = 0.0; /* Initialize the statistical counters. */ num_custs_delayed = 0; total_of_delays = 0.0; area_num_in_q = 0.0; area_server_status = 0.0; /* Initialize event list. Since no customers are present, the departure (service completion) event is eliminated from consideration. */ time_next_event[1] = sim_time + expon (mean_interarrival); time_next_event[2] = 1.0e+30; timing () /* Timing function. */ int i; double min_time_next_event = 1.0e+29; next_event_type = 0; /* Determine the event type of the next event to occur. */ for (i = 1; i <= num_events; ++i) if (time_next_event[i] < min_time_next_event) min_time_next_event = time_next_event[i]; next_event_type = i; 19

20 /* Check to see whether the event list is empty. */ if (next_event_type == 0) /* The event list is empty, so stop the simulation. */ printf ("\nevent list empty at time %f", sim_time); exit (1); /* The event list is not empty, so advance the simulation clock. */ sim_time = min_time_next_event; arrive () /* Arrival event function. */ double delay; /* Schedule next arrival. */ if(sim_time <= 480) /*Læt þetta ganga í 8 klst. eða 480 mín. */ time_next_event[1] = sim_time + expon (mean_interarrival); /* Check to see whether server is busy. */ if (server_status == BUSY) /* Server is busy, so increment number of customers in queue. */ ++num_in_q; /* Check to see whether an overflow condition exists. */ if (num_in_q > Q_LIMIT) /* The queue has overflowed, so stop the simulation. */ printf ("\noverflow of the array time_arrival at"); printf (" time %f", sim_time); exit (2); /* There is still room in the queue, so store the time of arrival of the arriving customer at the (new) end of time_arrival. */ time_arrival[num_in_q] = sim_time; else 20

21 (The affect /* Server is idle, so arriving customer has a delay of zero. following two statements are for program clarity and do not the results of the simulation.) */ delay = 0.0; total_of_delays += delay; /* Increment the number of customers delayed, and make server busy. */ ++num_custs_delayed; server_status = BUSY; /* Schedule a departure (service completion). */ time_next_event[2] = sim_time + expon (mean_service); else time_next_event[1] = 1.0e+30; /* Ef hermunartíminn er kominn í 480 mín viljum við ekki fá annan arrive atburð svo við setjum tímann á næsta arrive atburði = óendanlegt */ depart () /* Departure event function. */ int i; double delay; /* Check to see whether the queue is empty. */ if (num_in_q == 0) else /* The queue is empty so make the server idle and eliminate the departure (service completion) event from consideration. */ server_status = IDLE; time_next_event[2] = 1.0e+30; /* The queue is nonempty, so decrement the number of customers in queue. */ --num_in_q; 21

22 /* Compute the delay of the customer who is beginning service and update the total delay accumulator. */ delay = sim_time - time_arrival[1]; total_of_delays += delay; /* Increment the number of customers delayed, and schedule departure. */ ++num_custs_delayed; time_next_event[2] = sim_time + expon (mean_service); /* Move each customer in queue (if any) up one place. */ for (i = 1; i <= num_in_q; ++i) time_arrival[i] = time_arrival[i + 1]; report () /* Report generator function. */ /* Compute and write estimates of desired measures of performance. */ printf ("\n\naverage delay in queue%11.3f minutes\n\n", total_of_delays / num_custs_delayed); printf ("Average number in queue%10.3f\n\n", area_num_in_q / sim_time); printf ("Server utilization%15.3f\n\n", area_server_status / sim_time); printf ("Time simulation ended%12.3f minutes", sim_time); update_time_avg_stats () /* Update area accumulators for timeaverage statistics. */ double time_since_last_event; /* Compute time since last event, and update last-event-time marker. */ time_since_last_event = sim_time - time_last_event; time_last_event = sim_time; /* Update area under number-in-queue function. */ area_num_in_q += num_in_q * time_since_last_event; /* Update area under server-busy indicator function. */ area_server_status += server_status * time_since_last_event; 22

23 int main (int argc, char *argv[]) /* Main function. */ /* Read input parameters from command line arguments. */ if (5 == argc) /* convert input arguments to the double or integer */ mean_interarrival = atof (argv[1]); mean_service = atof (argv[2]); num_delays_required = atoi (argv[3]); SEED = atoi (argv[4]); else /* display some help to the user and exit */ printf ("usage: %s mean_interarrival mean_service num_delays_required SEED\n", argv[0]); printf ("example: %s \n", argv[0]); return 1; /* Specify the number of events for the timing function. */ num_events = 2; /* Write report heading and input parameters. */ printf ("Single-server queueing system\n\n"); printf ("Random number seed %u\n\n", SEED); printf ("Mean interarrival time%11.3f minutes\n\n", mean_interarrival); printf ("Mean service time%16.3f minutes\n\n", mean_service); printf ("Number of customers%14d\n\n", num_delays_required); /* Initialize the random number generator */ srand (SEED); /* Initialize the simulation. */ initialize (); /* Run the simulation while more delays are still needed. */ /*Nú er stöðvunarskilyrðið ekki lengur "num_custs_delayed < num_delays_required" heldur er hætt klukkan 17:00 og búið er að þjónusta alla viðskiptavini sem komu fyrir kl.17:00*/ int j = 1; while (j == 1) /* Determine the next event. */ 23

24 timing (); /* Update time-average statistical accumulators. */ update_time_avg_stats (); /* Invoke the appropriate event function. */ switch (next_event_type) case 1: arrive (); break; case 2: depart (); break; /*Viðbót*/ if ( sim_time > 480 && num_in_q == 0 && server_status == IDLE) j = 0; /* Invoke the report generator and end the simulation. */ report (); return 0; Output: Single-server queueing system Random number seed Mean interarrival time Mean service time minutes minutes Number of customers 1000 Average delay in queue minutes Average number in queue Server utilization Time simulation ended minutes 24

25 1.11 Náði ekki að leysa. Læt þó drög að lausn fylgja. Kóði: /* External definitions for single-server queueing system. GNU GCC compile ( gcc -o mm1.exe mm1tpr.c -lm GNU C standard format style ( indent --honour-newlines -nbbo --line-length120 foo.c -o bar.c */ #include <stdio.h> #include <stdlib.h> #include <math.h> #define Q_LIMIT 2 /* Limit on queue length. */ #define BUSY 1 /* Mnemonics for server's being busy */ #define IDLE 0 /* and idle. */ unsigned int SEED; int next_event_type, num_custs_delayed, num_delays_required, num_events, num_in_q, server_status, numbalk; double area_num_in_q, area_server_status, mean_interarrival, mean_service, sim_time, time_arrival[q_limit + 1], time_last_event, time_next_event[3], total_of_delays; double urand () /* Uniform variate generation function (0 1]. */ return ((1.0 + (double) rand ()) * (1.0 / ((double) RAND_MAX + 1.0))); double expon (double mean) /* Exponential variate generation function. */ /* Return an exponential random variate with mean "mean". */ return -mean * log (urand ()); initialize () /* Initialization function. */ /* Initialize the simulation clock. */ sim_time = 0.0; /* Initialize the state variables. */ server_status = IDLE; num_in_q = 0; time_last_event = 0.0; /* Initialize the statistical counters. */ 25

26 num_custs_delayed = 0; total_of_delays = 0.0; area_num_in_q = 0.0; area_server_status = 0.0; numbalk = 0.0; /* Initialize event list. Since no customers are present, the departure (service completion) event is eliminated from consideration. */ time_next_event[1] = sim_time + expon (mean_interarrival); time_next_event[2] = 1.0e+30; timing () /* Timing function. */ int i; double min_time_next_event = 1.0e+29; next_event_type = 0; /* Determine the event type of the next event to occur. */ for (i = 1; i <= num_events; ++i) if (time_next_event[i] < min_time_next_event) min_time_next_event = time_next_event[i]; next_event_type = i; /* Check to see whether the event list is empty. */ if (next_event_type == 0) /* The event list is empty, so stop the simulation. */ printf ("\nevent list empty at time %f", sim_time); exit (1); /* The event list is not empty, so advance the simulation clock. */ sim_time = min_time_next_event; arrive () /* Arrival event function. */ double delay; /* Schedule next arrival. */ time_next_event[1] = sim_time + expon (mean_interarrival); /* Check to see whether server is busy. */ if (server_status == BUSY) 26

27 else /* Server is busy, so increment number of customers in queue. */ ++num_in_q; /* Check to see whether an overflow condition exists. */ if (num_in_q > Q_LIMIT) ++numbalk; --num_in_q; /* There is still room in the queue, so store the time of arrival of the arriving customer at the (new) end of time_arrival. */ else time_arrival[num_in_q] = sim_time; /* Server is idle, so arriving customer has a delay of zero. (The following two statements are for program clarity and do not affect the results of the simulation.) */ delay = 0.0; total_of_delays += delay; /* Increment the number of customers delayed, and make server busy. */ ++num_custs_delayed; server_status = BUSY; /* Schedule a departure (service completion). */ time_next_event[2] = sim_time + expon (mean_service); depart () /* Departure event function. */ int i; double delay; /* Check to see whether the queue is empty. */ if (num_in_q == 0) /* The queue is empty so make the server idle and eliminate the departure (service completion) event from consideration. */ server_status = IDLE; 27

28 time_next_event[2] = 1.0e+30; else /* The queue is nonempty, so decrement the number of customers in queue. */ --num_in_q; /* Compute the delay of the customer who is beginning service and update the total delay accumulator. */ delay = sim_time - time_arrival[1]; total_of_delays += delay; /* Increment the number of customers delayed, and schedule departure. */ ++num_custs_delayed; time_next_event[2] = sim_time + expon (mean_service); /* Move each customer in queue (if any) up one place. */ for (i = 1; i <= num_in_q; ++i) time_arrival[i] = time_arrival[i + 1]; report () /* Report generator function. */ /* Compute and write estimates of desired measures of performance. */ printf ("\n\naverage delay in queue%11.3f minutes\n\n", total_of_delays / num_custs_delayed); printf ("Average number in queue%10.3f\n\n", area_num_in_q / sim_time); printf ("Server utilization%15.3f\n\n", area_server_status / sim_time); printf ("Number of balked customer\n\n", numbalk); /*Viðbót printf ("Time simulation ended%12.3f minutes", sim_time); update_time_avg_stats () /* Update area accumulators for time-average statistics. */ double time_since_last_event; /* Compute time since last event, and update last-event-time marker. */ time_since_last_event = sim_time - time_last_event; time_last_event = sim_time; /* Update area under number-in-queue function. */ area_num_in_q += num_in_q * time_since_last_event; /* Update area under server-busy indicator function. */ 28

29 area_server_status += server_status * time_since_last_event; int main (int argc, char *argv[]) /* Main function. */ /* Read input parameters from command line arguments. */ if (5 == argc) /* convert input arguments to the double or integer */ mean_interarrival = atof (argv[1]); mean_service = atof (argv[2]); num_delays_required = atoi (argv[3]); SEED = atoi (argv[4]); else /* display some help to the user and exit */ printf ("usage: %s mean_interarrival mean_service num_delays_required SEED\n", argv[0]); printf ("example: %s \n", argv[0]); return 1; /* Specify the number of events for the timing function. */ num_events = 2; /* Write report heading and input parameters. */ printf ("Single-server queueing system\n\n"); printf ("Random number seed %u\n\n", SEED); printf ("Mean interarrival time%11.3f minutes\n\n", mean_interarrival); printf ("Mean service time%16.3f minutes\n\n", mean_service); printf ("Number of customers%14d\n\n", num_delays_required); /* Initialize the random number generator */ srand (SEED); /* Initialize the simulation. */ initialize (); /* Run the simulation while more delays are still needed. */ while (sim_time <= 480) /* Determine the next event. */ timing (); /* Update time-average statistical accumulators. */ update_time_avg_stats (); 29

30 /* Invoke the appropriate event function. */ switch (next_event_type) case 1: arrive (); break; case 2: depart (); break; /* Invoke the report generator and end the simulation. */ report (); return 0; 30