A Proposal To Finding An Analytical Solution For The Probability Of j Units In An M/G/1 System

Transcription

1 A Proposal To Finding An Analytical Solution For The Probability Of j Units In An M/G/1 System By Philip Kalinda * MSc Business Analytics and Decision Sciences University of Leeds, Leeds, UK Introduction This proposal will attempt to derive an analytical solution for the probability of j units in an M/G/1/FCFS/ / (M/G/1) system by using the same method as for an M/M/1/FCFS/ / (M/M/1) system. The nature of the M/M/1 system allows the use a generic solution to determine the probability of j units in a queue, however, directly applying this solution to solve the probabilities for an M/G/1 system will not take into account the general service distribution of the system. With reference to the operating characteristics of both systems, the difference lies between the service characteristics. This paper will attempt to extract similarities from each set of operating characteristics in an attempt to produce a solution that can determine the probability of j units in an M/G/1 system taking into account the general service distribution. The systems being looked at will not take into account factors such as balking, reneging or jockeying within the calculations. Preliminaries Modelling Using Kendall s notation for queuing system, the paper will look at an M/M/1 and M/G/1 system. Looking at the M/M/1 system, this denotes that the system s arrivals follow a Poisson distribution with an expected rate λ per unit time, and that the time between successive arrivals has an exponential distribution, with mean value λ -1 units of time. An M/M/1 system uses random arrival patterns where units arrive to the system individually and independently. This notation also denotes that the service times of the system have an exponential distribution with the expected number of services per unit time being µ. This notation also denotes that there is only one server * Correspondence: Philip Kalinda, Studio 184, 23 King Street, Cambridge, CB1 1AH, UK. philipkalinda@gmail.com 1

2 in the system. This also indicates that the discipline is First Come First Serve (FCFS), that the capacity of the system is infinite and that the calling population is also infinite. The M/G/1 system used will have the same properties as that of the M/M/1 system except for the service characteristics. The service characteristic for the M/G/1 system follows a general service distribution. Notation FCFS is the queue discipline First Come First Serve. λ is the arrival rate per unit time. µ is the service rate per unit time. Pj is the probability there are j units in the system for j 1,2,3, P 0 is the probability there are no units in the system. P w is the probability an arriving unit has to wait. ρ is the service intensity of the system. is the average number of units in the queue of the system (M/M/1) is the average number of units in the queue of the M/M/1 system. (M/G/1) is the average number of units in the queue of the M/G/1 system. L is the average number of units in the system W q is the average time spent waiting in the queue W is the average time spent waiting in the system σ is the standard deviation Preliminary analysis When calculating the probability of j units in an M/M/1 system, P j, the formula used is Pj ( λ µ ) j P0 ρ j P0 For j 1,2,3,... 1 P0, the contents of the denominator represent the sum of a ρ ρ ρ... 1 ρ Geometric Progression with a ratio difference equal to ρ Hence P0 n 1 1 ρ + so that if n the value of P0 1 ρ and therefore, 2

3 Pj j (1 ρρ ) for j 1, 2, 3,... Other properties of an M/M/1 system are as follows: ρ 2 1 ρ λ 2 L + ρ ρ 1 ρ λ µ λ W q λ ρ µ λ λ W L λ W q + 1 µ 1 µ λ. Operating properties of an M/G/1 system are as follows: P 0 1 λ µ (λσ ) 2 + ( λ µ )2 2(1 λ µ ) L + λ µλ 2 σ 2 λ 2 µ µ + 2λ 2(µ λ) W q L (λσ ) 2 + ( λ q λ µ )2 2λ(1 λ µ ) W W q + 1 µ L µλ 2 σ 2 λ 2 λ µ + 2λ 2λ(µ λ) 3

4 Reconstruction and Proposal Looking at the operating characteristics of both systems we are able to extricate similarities between the general solutions. Now looking at the M/M/1 system, the average length of the queue,, can be calculated with the service intensity of the system ( ρ ) with the formula ρ 2 1 ρ. This can be restructured to incorporate the probability that there are no units in the system (P 0 ). ρ 2 Therefore, P 0 Where P 0 1 ρ. P 0 ρ ( λ 2 µ )2. The difference between the two operating characteristics lies in their formulas. Therefore, incorporating the element in the P j formula should provide a solution that could be applicable to both systems. Knowing that Pj ( λ µ ) j P0 ρ j P0, we can insert our restructured formula for P 0 into this the P j formula to give: Pj P0( λ ( λ µ ) j µ )2 ( )( λ µ ) j. Expanding this formula using (M/M/1) where λ 2, we confirm that Pj ( ( λ µ )2 )( λ µ ) j with the following working; 4

5 ( λ µ )2 ( λ µ ) j ( λ 2 ( λ µ )2 ( λ µ ) j () ) ( ) λ 2 ( λ µ )2 ( λ µ ) j (µ 2 µλ) ( λ µ )2 ( λ µ ) j µ 2 ( λ µ )2 ( λ µ ) j µλ ( ) ( ) λ 2 λ 2 ( λ µ )2 ( λ µ ) j µ 2 ( λ µ )2 ( λ µ ) j µλ ( 1 λ 2 λ 2 µ )(λ 2 µ ) j µ 2 ( λ µ )(λ 2 µ ) j µ ( λ µ ) j ( λ µ )(λ µ ) j (1 ( λ µ ))(λ µ ) j (1 ρ)ρ j Where ρ λ µ With this approach, we are able to calculate the value of P j with the use of P 0 and (M/M/1). Using the same approach, (M/G/1) is then inserted into the formula Pj P0( λ ( λ µ ) j µ )2 ( )( λ µ ) j to determine the probability there are j units in the M/G/1 system. ( λ µ )2 ( λ µ ) j ( λ µ )2 (2(1 λ µ ))(λ µ ) j ( (λσ ) 2 + ( λ ) µ )2 (λσ ) 2 + ( λ µ )2 2(1 λ µ ) 2( λ µ )2 (1 λ µ )(λ µ ) j (2λ 2 2 λ 3 µ )(λ µ ) j (λσ ) 2 + ( λ µ 2 λ 2 σ 2 + λ 2 µ )2 (2 2 λ µ )(λ µ ) j µ 2 σ (1 λ µ )(λ µ ) j µ 2 σ

6 2P 0 ( λ µ ) j µ 2 σ 2 +1 This formula uses the same approach as the M/M/1 system to calculate P j, however, to be applicable to the M/G/1 system, (M/G/1) has been inserted into the P j formula as to take into account the general service distribution, more specifically the time variation σ. 6