AN M/M/1 QUEUEING SYSTEM WITH INTERRUPTED ARRIVAL STREAM K.C. Madan, Department of Statistics, College of Science, Yarmouk University, Irbid, Jordan

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1 REVISTA INVESTIGACION OPERACIONAL Vol., No., AN M/M/ QUEUEING SYSTEM WITH INTERRUPTED ARRIVAL STREAM K.C. Maan, Department of Statistics, College of Science, Yarmouk University, Irbi, Joran ABSTRACT We stuy an M/M/ queueing system in which the arrival stream is 'shut off' from time to time. The timeepenent probability generating functions of the number in the system have been foun. The corresponing steay state results have been erive. The mean number in the system as well as server's utilization time an ile time have been obtaine explicity. In a particular case, some wellknown results have been erive. Key Wors: interrupte arrival stream, variable batch arrivals, exponential service, probability generating function, steay state, utilization time, ile time, mean number in the system. RESUMEN Estuiaremos un sistema e colas M/M/ en el que la corriente e arribos se esembaraza e vez en vez. Han sio hallaas las funciones generatrices e probabiliaes epenientes el tiempo. Los resultaos corresponientes para el estao previo han sio erivaos. El número promeio en el sistema así como el tiempo e la utilización el servior y el tiempo ocioso son obtenios explícítamente. Algunos resultaos bien conocios son erivaos para un caso particular. MSC 6K5.. INTRODUCTION Queueing systems with vacations or service interruptions or breakowns have been stuie by numerous authors incluing Keilson an Servi [986], Scholl an Kleinrock [983], Cramer [989], Shanthikumar [988], Doshi [986] an Maan [99, 995]. In this paper we investigate a queueing system in which there are no server vacations or breakowns. Instea, the arrivals into the system are interrupte in the sense that the arrival stream gets 'shut off' from time to time. Such a situation coul have efinite effect on queue characteristics incluing queue length an server's utilization time. Some examples where such a situation coul be encountere are the following. Aircrafts may stop laning at an airport for some time ue to ba weather conitions. Branches of an assembly line coul stop from time to time thereby blocking the input to the next branch. Flow of oil into or out of a refinery or flow of water into or out of a reservoir coul experience ranom stoppages ue to some reasons. Similarly flow of vehicles in a large network of a traffic system coul stop for some time an then may become normal again an so on.. THE MATHEMATICAL MODEL Customers arrive at the system in batches of variable size. Given that the arrival stream (input source) is 'on' at time t, let λc i t (λ > ) be the first orer probability that a batch size i customers arrives uring the time interval (t, t+t) where < c i < an =. c i Customers are serve one by one by the server with their service times having negative exponential istribution function B(x) = - e- μt, t >, where μ (μ > ) is the mean service time. Given that the arrival stream is 'on' at time t, let ξt (ξ > ) be the first orer probability that the arrival stream will be 'off' uring the short interval of time (t, t+t). 35

2 Given that the arrival stream is 'on' at time t, let t ( > ) be the first orer probability that the arrival stream will be 'on' uring the short interval of time (t, t+t). Further, we assume that the various stochastic processes involve in the system are inepenent of each other. 3. DEFINITIONS AND SYSTEM EQUATIONS Let p n (t), ( n ) be the probability that at time t there are n units in the system incluing one in service, if any, an the input source is 'on' an let q n (t), (n ) be the probability that at time t there are n units in the system incluing one in service, if any an the input source is 'off'. The system is governe by the following set of forwar ifferential-ifference equations n pn (t) + (λ + μ + ξ)p n (t) = t λcip n-i (t) + μp n+ (t) + q n (t) (n ) () p (t) + (λ + ξ)p (t) = μp (t) + q (t) () t qn (t) + (μ + )q n (t) = μq n+ (t) +ξp n (t) (n ) (3) t q (t) + q (t) = μq (t) +ξp (t) (4) t 4. THE GENERATING FUNCTIONS OF THE NUMBER IN THE SYSTEM We efine p(z,t) = pn(t)z n an q(z,t) = qn(t)z n as the probability generating functions of the numbers in the system when the input source is 'on' an 'off' respectively. Let c(z) = probability generating function for the sequence ci of arrivals. i c z enote the i Let us assume that the system starts when the input source is 'on' an there are j units in the system, so that the initial conition is p n () = δ n,j where δ nj is the Kronecker's elta. (5) Taking Laplace transform of equations () through (4) an using initial conition (5), we have n (s + λ + μ + ξ)p n (s) = δ n,j + λcip n-i (s) + μp n+ (s) + q n (s) (n ) (6) (s + λ +ξ)p (s) = μp (s) + q (s) (7) (s + μ + )q n (s) = μq n+ (s) + ξp n (s) (n ) (8) (s + )q (s) = μq (s) + ξp (s) (9) Multiplying equations (6) through (9) by suitable powers of z yiels 36

3 [(s + λ + μ + ξ)z - μ - λc(z)z]p(z,s) = z j+ + z - )p + zq(z,s) () [(s + μ + )z - μ]q(z) = ξzp(z,s) + z - )q (s) () Using equation () in (), we obtain p(z,s) = [z j+ + z )p (s)][s + μ + )z μ] + μz(z )q (s) [(s + μ + )z μ][(s + λ + μ + ξ)z μ λc(z)z] ξz () Further, equation () can be written as ξzp(z, s) + z )q (s) (z,s) = (s + μ + )z μ q (3) Now for z =, equations () an (3) respectively yiel s + s + p(,s) = = (s + )(s + ξ) ξ s(s + (4) ξp(,s) ξ q(,s) = = (s + ) s(s + (5) We note that p(,s) + q(,s) = s as it shoul be. Inverting the transforms (4) an (5), we obtain the respective probabilities that at time t the input source is 'on' an 'off' as follows ξ ( ξ+)t p(t) = = e (6) ξ + ξ + ξ ξ ( ξ+)t q(t) = = e (7) ξ + ξ + Obviously, p(t) + q(t) = as it shoul be. The enominator of the right han sie of equation () has zeroes insie the unit circle z =. (For a proof, one can use Rouche's theorem, see Maan [4]). Let these two zeroes be esignate as z an z. Then since p(z,s) is regular insie z =, the numerator of the right han sie of equation () must vanish for each of these zeroes, thus yieling the following two equations j [ z + + z - )p (s)] [(s + μ + )z - μ] + μz (z - )q (s) = (8) j [ z + + z - )p (s)] [(s + μ + )z - μ] + μz (z - )q (s) = (9) The unknowns p (s) an q (s) can be etermine by solving equations (8) an (9) simultaneously. Thus p(z) an q(z) can be completely etermine. 37

4 5. STEADY STATE SOLUTION Let p n an q n efine the steay state probabilities corresponing to the time epenent probabilities p n (t) an q n (t) efine above an let p(z) an q(z) enote the respective steay state probability generating functions for the number in the system. Then to erive the steay state results we apply the well-known Tauberian property Lim sf(s) = Lim f(t) to equations () through (5) an obtain s t p(z) = q(z) = z )[ z ) + z]p [ z ) + z][ λ + μ + ξ)z μ λc(z)z] ξz ξzp(z) + z )q z ) + z + μz(z )q () () At z = equations () is ineterminate of the / form. Therefore, we use L'Hopital's rule to fin limit of p(z) as z. Thus we have p() = μ( p + q ) λe(i) () q() = μξ( p + q ) λe(i) μ ( with λe(i) < (3) where c'() = ic i = E(I) is the mean batch size of arriving customers. Equations () an (3) respectively give the probabilities that the arrival stream is 'on' or 'off '. Since p() + q() =, aing () an (3), we obtain p + q = λe(i) μ ( with λe(i) < (4) Since equation (4) gives the probability that the server is ile, no matter whether the arrival steam is 'on' or 'off ', we can fin its complement to get the system's utilization factor as λe(i) ρ = (5) We will continue the analysis for the case of single arrivals. In that case c = an c i = for i. Consequently c(z) = z. Then the enominator of the right sie of equation () can be written as [-λ(μ + )z 3 + (μ + λμ + λ + μ + μξ)z - λ + ξ + + μ)z + μ ] which can be factore as (z-) [-λ(μ + ) z + λ + μ + z - μ ] so that now the factor (z - ) can be cancele out with that of the numerator. Therefore, the expression in () can be re-written as p(z) = μ[ z ) + z]p λ( μ + )z + μzq + λ + μ + z u (6) 38

5 Now, the enominator of the right han sie of equation (6) has two zeroes given by λ + μ + m μ ( λ + μ + 4λ( μ + ). One of these zeroes is insie the unit circle z =. Let this λ( μ + ) zero be enotes as z*. The enominator of the right han sie of (6) must vanish for this zero, giving [ μ ( μ + )z*]p = (7) z * q Since in the case of single arrivals E(I) =, equation (4) becomes λ + q = where λ < p μ ( (8) p Solving (7) an (8), we obtain [ λ] z * = (9) μ ( ( z*) which is the steay state probability that the server is ile, even though the arrival stream is 'on'. λ] μ ( μ + )z * q = (3) z*) which is the probability that the server is ile when the arrival stream is 'off '. 6. THE MEAN NUMBER IN THE SYSTEM Let r(z) = p(z) + q(z) be the probability generating function of the number in the system no matter whether the arrival stream is 'on' or 'off ', where p(z) an q(z) are given by equations (6) an () respectively. Then the mean number, L, in the system is given by L = r(z) at z =. Carrying out the computations an z simplifying, we have ( λ( μ + ) μ ) μq + ξμ + λ )p ξ ξμ L = + (3) ( λ) ( 7. A PARTICULAR CASE If we assume ξ =, which means that there are no interrumptions in the arrival stream, then the enominator of the right sie of (6) becomes - λ(μ + )z + λ + μ +) - μ. It is easy to see that one zero of μ μ this polynomial which is insie the unit circle z = is. Setting z* = in equation (3) we get q = μ + μ + λ as it shoul be an the (9) gives p = - which is a well-known result of the queueing system M/M/. μ Further, with ξ =, q = an p = - μ λ, equation (6) becomes 39

6 λ μ[ z ) z] p(z) μ = (3) λ ( μ + )z + λ + μ + )z μ In we ivie the numerator an the enominator of the right sie of equation (3) by an take limit as an simplify, we obtain λ μ p(z) μ = (33) μ λz which is again a well-known result giving the probability generating function of the number in the M/M/ queueing system. Again, letting ξ = an using q = an p = - μ λ in equation (3), we have the mean queue length in this particular case as L = λ μ λ which is also a well-known result. (34) REFERENCES CRAMER, M. (989): "Stationary istributions in a queueing system with vacation times an limite service", Queueing Systems Theory Appli. 4(), DOSHI, B.T. (986): "Queueing systems with vacation - a survey", Queueing Systems, KEILSON, J. an L.D. SERVI (986): "Oscillating ranom walk moels for GI/G/ vacation systems with Bernoulli Scheules", Journal of Applie Probability, 3, MADAN, K.C. (99): "An M/G/ queueing system with compulsory server vacations", Trabajos e Investigación Operativa 7(), 5-5. (995): "A bulk input queue with a stan by", South African Statistical Journal 9(), -7. SHANTHIKUMAR, G.J. (988): "On stochastic ecomposition in the M/G/ queue with generalize vacation", Oper. Res. 36, SCHOLL, M. an L. KLEINROCK (983): "On the M/G/ queue with rest perios an certain service inepenent queueing systems", Operations Research 36,