INTERNATIONAL JOURNAL OF MATHEMATICAL SCIENCES AND APPLICATIONS Volume 5 Number 1 Jauary-Jue 215 ICC WORLD TWENTY2-214 ( WORLD CUP-214 )- A CASE STUDY Bhavi Patel 1 ad Pravi Bhathawala 2 1 Assistat Professor, Humaities Departmet, Sakalchad Patel College of Egieerig, Visagar, Gujarat Idia 2 Professor of Mathematics, Auro Uiversity, Opp. ONGC, Hajira Road, Surat, Gujarat,Idia Abstract: The preset paper describes the pheomea of the Twety2 matches. I Twety2 match, two opeig batsme ope the iigs. From the time duratio for which they are playig, we ca fid the service rate ad the arrival rate for the pitch which serves the batsme. Other batsme are waitig for their tur. So, we apply queuig model M/M/1 to Twety2 matches ad determie the probability of wi by first iigs ad secod iigs i Twety2 match. Keywords: Twety2 match, queue, queuig model, game of cricket HISTORY OF TWENTY-2 CRICKET: Twety 2 cricket, ofte abbreviated to T2, is a form of cricket origially itroduced i Eglad by the Eglad ad Wales Cricket Board (ECB) i 23. A Twety 2 game of cricket ivolves two teams, each has a sigle iigs, battig for a maximum of 2 overs. A Twety2 game of cricket is completed i about three hours, with each iigs lastig aroud 75 9 miutes (with a 1 2 miutes iterval ) thus brigig the popularity to the game of cricket. It was itroduced to attract the spectators at the groud ad viewers o tele-visio ad as such it has bee very successful game of cricket. Sice its iceptio the game has spread aroud the cricket world. It is played betwee the iteratioal teams that play game of cricket. O most iteratioal tours there is at least oe Twety2 match. The iaugural ICC World Twety 2 was played i South-Africa i 27 with Idia wiig by five rus agaist Pakista i the fial. Curretly, Sri-laka are the reigig champios after wiig the ICC World Twety 2-214. INTRODUCTION As i the other two formats, Tests ad ODIs, i Twety2 match, two opeig batsme ope the iigs i each iigs. This pair of batsme is served by a sigle server i.e. cricket pitch. Durig the time period for which these two are playig, the oe dow batsma is waitig i a dressig room i.e. waitig i a queue. By same way, other batsme are also waitig for their tur. From the duratio of time for opeig batsme, we ca derive the service rate ad the arrival rate for the cricket pitch. The time for which oe pair of batsme is battig, is take as the service time. As soo as, the wicket falls, the ew batsma arrives at the pitch. So, the arrival time of the waitig customers (cricketers) is same as the service time. The customers are beig served by a sigle server i.e. the battig pitch. The service time is radom, so the arrival time is also radom. Thus, we ca apply the M / M /1 queuig model to the Twety 2 match. The utilizatio factor for the pitch is take as = 1. M / M /1QUEUING MODEL I this model, there is a limit o the umber i the system (maximum queue legth = N 1). Example iclude a oe lae drive-i widow i a fast-food restaurat ad a iigs i a Twety2 match i cricket.
8 BHAVIN PATEL AND PRAVIN BHATHAWALA I these two examples, whe the umber of customers i the system is reached to N, o more arrivals are allowed i the system. Thus, we have,,,1, 2,..., N 1, N, N 1,...,,1, 2,... Usig, the probability of customers i the system is p p, N, N The value of is determied from the equatio p 1, which gives p (1 2. N ) 1 or p 1, 1 N 1 1 1, 1 N 1 Thus, (1 ), 1 N 1 1 p,,1,..., N 1, 1 N 1 The value of eed ot be less tha 1 i the case of Twety2 match, because arrivals at the system are cotrolled by the system limit N. This meas that eff is the rate which matters i this case. Because customers will be lost whe there are N i the system, p eff lost pn (1 pn ). I the case of Twety2 match, eff.. Sice 1, eff The expected umber of customers i the system is determied by lost N
ICC WORLD TWENTY-2, 214 (WORLD CUP-214)- A CASE STUDY 9 p L p N s ( 2 2 3 3... N ) N p (1 2 3... N ) 1 N( N 1) N 1 2 N LS. 2 EXPECTATION IN TWENTY2 MATCH: I Twety2 match, utilizatio factor = 1 ad the umber of customers (wickets) fallig is N =1 i each iigs. Therefore, the expected umber of customers (wickets) i each iigs or expectatio of customers (wickets) is 1 5. 2 That meas i each iigs, o average 5 wickets are fallig. It may vary earer 5 to i.e. 4 or 6 wickets i each iigs. We applied queuig model to Twety2 cricket i particular to the ICC World Twety2-214. As a result, firstly we got the expected umber of wickets durig World Cup-214. The actual record of the World Cup-214 is as uder: Match No. Of Wkts No. Of Wkts Average Match No. of Wkts No. Of Wkts Average Sr. No. i 1st I. i 2d I. No. Of Wkts Sr. No. i 1st I. i 2d I. No. Of Wkts 1 7 3 5 13 4 4 4 2 7 8 7.5 14 5 7 6 3 6 1 3.5 15 5 7 6 4 5 1 7.5 16 7 1 8.5 5 7 3 5 17 5 1 7.5 6 6 8 7 18 1 1 1 7 1 1 5.5 19 5 3 4 8 7 1 8.5 2 6 1 8 9 9 1 9.5 21 6 4 5 1 4 4 4 22 4 4 4 11 8 4 6 23 4 4 4 12 7 2 4.5 Total: 23 144 137 14.5 No. = umber Wkts = Wickets I. = Iigs Actual Expectatio is 14.5 6.1 6 23 wickets. This data shows that, durig ICC World Twety2-214, the average umber of wickets that has bee falle is 6. This average is close to the ideal average 5. This is the real applicatio of queuig models i the real world.
1 BHAVIN PATEL AND PRAVIN BHATHAWALA CALCULATION OF PROBABILITIES IN TWENTY2 MATCH Secodly, we seek to fid the probability that the team battig first will wi a Twety2 match ad the probability that the team battig secod will wi. We eglect the tied Twety2 matches, ulike ODIs, because i that case, the super over comes i the frame ad gives the result of the match. For this, we defie the evets of team battig first ad team battig secod. At the start of the game, the coi is tossed to decide which team will bat first. Let B 1 be the evet that the team battig first ad B 2 be the evet that the team battig secod. 1 P( B ) P( B ). 2 1 2 Let I be the evet that the 1 st iigs is completed, irrespective of the umber of wickets fallig i the iigs. P( I ) P( 1 i the 1 st iigs) 1 1 1 (11 times ) 11 11 11 P(I) = 1.. Now, Let I 1 be the evet that the team battig first wi ad I 2 be the evet that the team battig secod wi. The, the probability of I 1 is determied by P( I 1) The probability of I 2 is determied by = P(B 1 )P(I) + P(B 2 ) P( = 1 i the 2 d iigs) = 1 (1) 1 1 1 1 12 2 2 11 2 22 22 d P(I 2 ) = P( B2 ) P( 9 i the 2 iigs) = 1 1 1... 1 (1 times) 2 11 11 11 = 1 1 1. 2 11 22 This meas that, out of Twety2 matches 12 matches are wo by team battig first ad 1 matches are wo by team battig secod. This ca be verified from the data of the ICC World Twety2-214. The actual record is as uder: Match Sr. No. 1st I. Wi 2d I. Wi Match Sr. No. 1st I. Wi 2d I. Wi 1 L 1 W 13 L 1 W 2 1 W L 14 1 W L 3 1 W L 15 1 W L 4 1 W L 16 1 W L 5 L 1 W 17 1 W L 6 1 W L 18 1 W L 7 L 1 W 19 L 1 W Table Cot d
ICC WORLD TWENTY-2, 214 (WORLD CUP-214)- A CASE STUDY 11 8 1 W L 2 1 W L 9 1 W L 21 1 W L 1 L 1 W 22 L 1 W 11 L 1 W 23 L 1 W 12 L 1 W Total:23 13 W 1 W meas Lost(L) ad 1 meas Wo(W) The above table shows that durig the ICC World Twety2-214, out of 23 Twety2 matches 13 matches have bee wo by team battig first ad 1 have bee wo by team battig secod. We observe that the theoretical probabilities match with the actual probabilities. CONCLUSION We coclude that, i T2 game of cricket, out of 22 Twety2 matches12 matches are wo by team battig first ad 1 matches are wo by team battig secod. That is, the probability that team battig first wi is 12.55 22 ad the probability that team battig secod wi is 1.45 22. I other words, it meas that out of 1 Twety2 matches 55 matches are wo by team battig first ad matches are wo by team battig secod. Durig ICC World Twety2-214, the probability that team battig first wi is 13.57 23 ad the probability that team battig secod wi is 13.43 23. This shows that the theoretical probabilities match with the actual probabilities. Moreover, durig ICC World Twety2-214, the actual expectatio of wickets fallig is wickets. REFERENCES [1] H.A. Taha, Operatios Research-A Itroductio. 8 th Editio, ISBN 13188923. Pearso Educatio, 27. [2] J.D.C. Little, A Proof for the Queuig Formula:, Operatios Research, 9(3), 1961, pp. 383-387, doi:1.237/ 16757. [3] Cooper RB (1972). Itroductio to Queuig Theory. McMilla: New York. [4] Tijms HC (1986). Stochastic Modellig ad Aalysis. A Computatioal Approach. Wiley: Chichester. [5] K. Rust, Usig Little s Law to Estimate Cycle Time ad Cost, Proceedigs of the 28 Witer Simulatio Coferece, IEEE Press, Dec. 28, doi:1.119/wsc.28.4736323. [6] Worthigto D ad Wall A (1999). Usig the discrete time modellig approach to evaluate the time-depedet behaviour of queuig systems. J Opl Res Soc 5: 777-888. [7] Steveso WJ (1996). Productio/Operatios Maagemet, 6 th ed. Irwi, McGraw-Hill, USA.