ELEMENTARY MARKOV QUEUEING SYSTEMS WITH UNRELIABLE SERVER

Transcription

1 Nuber 4. Volue IV. Deceber 2009 EMENTARY MARKOV QUEUEING SYSTEMS WITH UNRIABLE SERVER Michal Dorda Suary:This paper presents soe exaples of eleentary Marov unreliable queueing systes and their atheatical odels. Selected odels are represented by a state transition diagra and by appropriate syste of linear equations for steady state probabilities coputation. Key words: Marov queueing syste M/M// unreliable server.. INTRODUCTION In the queueing theory we usually ignore the fact that utility server is often a technical device and its breadown can occur. Coon analytic odels (see for exaple in [] or [2]) assue that the server wors reliably without failures causing its deficit. If failures of the server occur often and we would use an analytic odel of reliable queueing syste for odeling studied syste reached outcoes can be inaccurate. In this case the suitable odel of the unreliable queueing syste should be used. This paper presents soe exaples of eleentary Marov unreliable queueing systes odels. Selected odels are represented by a state transition diagra and by appropriate syste of linear equations for steady state probabilities coputation. Steady state probabilities are needed for perforance easures coputation. More inforation about entioned odels can be found in [3]. 2. COMMON ASSUMTIONS Let us assue the queueing syste with a single unreliable server. Incoing custoers wait for the service in a queue with capacity of - custoers. Thus there are places in the syste. Custoers coe to the syste according to a oisson process with a rate λ that eans interarrival ties are exponentially distributed with ean value equal to λ. Failures occur according to the oisson process too but with rate λ. The tie between failures is an exponential rando variable with ean value equal to λ. Costuer service ties and ties between failures are exponential rando variables with paraeters μ and μ respectively. Thus ean costuer service tie is equal to μ and ean server repair tie is μ. Custoers are served one by one according to FIFO (First In - Ing. Michal Dorda VŠB - Technical University of Ostrava Faculty of Mechanical Engineering Institute of Transport 7. listopadu Ostrava-oruba Tel.: E-ail: ichal.dorda@vsb.cz Dorda - Eleentary Marov queueing systes with unreliable server 69

2 Nuber 4. Volue IV. Deceber 2009 First Out) discipline. On the basis of these assuptions we can say that all presented systes are according to Kendall s notation M/M// systes with an unreliable server. 3. UNRIABLE M/M// QUEUEING SYSTEMS WITH CUSTOMER SERVICE REEATING Let us consider that a server breadown can occur at any tie. That eans the server can brea if it is busy (a costuer is in the service) or idle (there is no custoer in the service). After occurrence of breadown two different events can occur (if there is a costuer in the service ust at the oent): if there is less than - custoers in the queue costuer coes bac to the first queue place and after server repair his service starts fro the beginning if there is exactly - custoers in the queue costuer leaves the syste and we consider he is reected. Let illustrate this queueing odel graphically as a state transition diagra (see in fig. ). To distinguish all presented odels let us designate this odel as st odel. Vertices represent particular syste states and oriented edges indicate possible transitions with corresponding rate. Notice that graph in fig. is drawn without loops. Fig. - State transition diagra of st odel (without loops). States of st odel we can divide to two groups: states denoted as 0 where the first sybol 0 says that server is failure free and the second sybol { 0...} represents nuber of the costuers in the syste states denoted as where the first sybol eans server breadown and the second 0... is nuber of the waiting custoers. sybol { } On the basis of the state transition diagra we can obtain finite syste of the differential equations for probabilities of the particular states depending on tie t. For t we get the syste of the linear equations for steady state probabilities that are not dependent on tie t. For st odel we obtain 2 linear equations: ( λ λ ) 00 μ0 μ 0 0 ( λ μ λ ) 0 μ0 0 = μ for = 2... Dorda - Eleentary Marov queueing systes with unreliable server 70

3 Nuber 4. Volue IV. Deceber 2009 ( μ ) 0 λ 0 00 ( λ μ ) 0 λ ( λ ) μ for = λ 0 λ 2 μ plus equation 0 =. = 0 = 0 By solving of this linear equations syste we get stationary probabilities of the particular syste states that are needed for perforance easures coputing of studied queueing syste. Let consider two selected perforance easures - ean nuber of the custoers in the service and ean nuber of the waiting custoers. For st odel we get: = 0 and = ( ) 0 = 2 =. Now let us focus on the siple odification of st odel. Assue that the server breadown can occur ust during costuer service. If the server is idle then there is zero probability of the server failure. The state transition diagra of this odel (denoted as 2 nd odel) can be seen in fig. 2. Notice that states representation is the sae with st odel. Fig. 2 - State transition diagra of 2 nd odel (without loops). The ost rearable difference is that in fig. 2 there is not the state denoted as 0. It is the consequence of our assuption that breadown can not occur if the server is idle. For the 2 nd odel we obtain for t 2 linear equations: μ 00 0 ( λ μ λ ) 0 μ0 ( μ ) 0 0 μ for = 2... λ 0 0 ( λ μ ) λ ( λ ) μ for = Dorda - Eleentary Marov queueing systes with unreliable server 7

4 Nuber 4. Volue IV. Deceber λ 0 λ 2 μ including equation 0 =. = 0 = In this case for selected perforance easures we get: = 0 and = ( ) 0 = 2 =. 4. UNRIABLE M/M// QUEUEING SYSTEMS WITH MASS CUSTOMERS DEARTURE Let assue that server breadown occurrence causes departure of all costuers fro the syste (these costuers are reected). Further we assue that incoing custoers are reected until server is not repaired. Let us consider two versions of studied queueing syste. In the first case (3 rd odel) breadown ay occur at any tie in the second case (4 th odel) ust during the costuer service. The state transition diagras are shown in fig. 3 and 4 respectively. Fig. 3 - State transition diagra of 3 rd odel. Dorda - Eleentary Marov queueing systes with unreliable server 72

5 Nuber 4. Volue IV. Deceber 2009 Fig. 4 - State transition diagra of 4 th odel. The states of both odels can be interpreted as: state denotes server breadown there is no custoer in the syste and incoing custoers are reected states 0... correspond to states of reliable M/M// queueing syste. On the basis of the state transition diagra in fig. 4 we can see that in 4 th odel there is no transition fro the state 0 to the state (breadown can not occur when the server is idle). For 3 rd odel we obtain 2 linear equations: 0 = μ p λ = 0 ( λ λ ) 0 μ ( λ μ λ ) μ ( μ λ ) 0 = μ p 0 = λ for = λ = with equation p =. = 0 For selected perforance easures we can write: = and = ( ) = 2. For 4 th odel we can write 2 linear equations too: 0 = μ p λ 0 = 0 = μ λ μ p ( λ μ λ ) μ ( μ λ ) 0 = 0 λ for = 2... = λ Dorda - Eleentary Marov queueing systes with unreliable server 73

6 Nuber 4. Volue IV. Deceber 2009 plus equation p =. = 0 In this case for selected perforance easures we get: = and = ( ) = UNRIABLE M/M// QUEUEING SYSTEM WITH COSTUMER SERVICE COMLETING AFTER OCCURRENCE OF BREAKDOWN Server failures in all the previous odels brea the costuer service iediately. But in soe cases occurrence of breadown do not cause an iediate interruption of the custoer service. Loo at the appropriate queueing odel. At first let assue that the probability distribution of the custoer service tie before and after occurrence of breadown is the sae - an exponential distribution with paraeter μ. The state transition diagra of this queueing odel (5 th odel) is shown in fig. 5. Fig. 5 - State transition diagra of 5 th odel. The states of 5 th odel can be described as: states 0... correspond to states of reliable M/M// queueing syste states denoted as where sybol { 2...} describes nuber of custoers in the syste (a costuer in the service and - waiting custoers) and sybol says that server breadown have occurred states denoted as where sybol indicates server repairing and sybol { 2... } describes nuber of waiting custoers. Dorda - Eleentary Marov queueing systes with unreliable server 74

7 Nuber 4. Volue IV. Deceber 2009 For 5 th odel we get for t 3 linear equations: ( λ λ ) 0 μ μ 0 ( λ μ λ ) μ ( μ λ ) 0 = μ for = λ = ( λ ) λ ( λ ) μ μ for = μ λ ( λ μ ) 0 ( λ ) 0 λ 0 μ = 0 = μ λ μ for = = μ λ 2 μ with equation =. = 0 = = 0 Selected perforance easures can be coputed as: = = [ ] and = ( ) ( ) = 2. = 6. EXECUTED EXERIMENTS AND THEIR OUTCOM Let consider unreliable M/M//5 syste (a axial queue length is 4 costuers). Let consider 3 constant syste paraeters - λ = 9h μ = 0h and μ = 02h. The last paraeter λ will be increased fro the iniu value λ = 0000h (this value corresponds to ean tie between failures equal to0 000h ) to axiu value λ = 0 h (ean tie between failures is 0 h ). λ Establish ρ = as breadowns load. Focus on dependence of selected perforance μ easures on ρ. These graphical relations we can see in fig. 6 up to fig. 0. Outcoes obtained by analytic coputation are suppleented by outcoes gained by siulation of studied syste in software Witness. All siulation experients were executed for 2 years of real tie. The constant curve shown in all graphs reflects reliable syste perforance easure behavior. Dorda - Eleentary Marov queueing systes with unreliable server 75

8 Nuber 4. Volue IV. Deceber 2009 Reliable syste Coputation Siulation Reliable syste Coputation Siulation Fig. 6: Dependence of on ρ (on the left) and on ρ (on the right) - st odel. Reliable syste Coputation Siulation Reliable syste Coputation Siulation E-04 63E-04 83E-04 3E-03 25E-03 56E-03 7E-03 0E-02 7E-02 28E-02 36E-02 50E-02 83E-02 25E-0 Fig. 7: Dependence of on ρ (on the left) and on ρ (on the right) - 2 nd odel. Reliable syste Coputation Siulation Reliable syste Coputation Siulation Fig. 8: Dependence of on ρ (on the left) and on ρ (on the right) - 3 rd odel. Reliable syste Coputation Siulation Reliable syste Coputation Siulation Fig. 9: Dependence of on ρ (on the left) and on ρ (on the right) - 4 th odel. Dorda - Eleentary Marov queueing systes with unreliable server 76

9 Nuber 4. Volue IV. Deceber 2009 Reliable syste Coputation Siulation Reliable syste Coputation Siulation Fig. 0: Dependence of on ρ (on the left) and on ρ (on the right) - 5 th odel. On the basis of the coparison of the gained dependence for particular odels we can clai that there are no significant differences between analytic and siulation outcoes. 7. CONCLUSION The presented paper introduces soe analytic odels of Marov unreliable queueing systes. All odels are analytic solved through the state transition diagra and the syste of the linear equations describing syste of behavior in steady state. On the basis of stationary probabilities nowledge perforance easures can be coputed. In this paper two perforance easures - ean nuber of the custoers in the service and ean nuber of the waiting custoers - were considered. Created analytic odels were validated by siulation experients in Witness. Executed experients do not reveal significant differences between analytic and siulation outcoes. REFERENC [] KLUVÁNEK. - BRANDALÍK F. Operační analýza I (Teorie hroadné obsluhy). Bratislava: ALFA Bratislava [2] EŠKO Š. Operačná analýza 2 [online]. Žilina: Žilinsá univerzita s. [cit ] Dostupné z : < [3] DORDA M. Modelování spolehlivosti systéů hroadné obsluhy v dopravě: disertační práce. Ostrava: VŠB - TU Ostrava Faulta stroní s. Reviewers: doc. Ing. Josef Vole CSc. University of ardubice Jan erner Transport Faculty Departent of inforatics in Transport doc. RNDR. Antonín Tuzar CSc. Czech Technical University in rague Faculty of Transportation Sciences Departent of Logistic and Transportation rocesses Dorda - Eleentary Marov queueing systes with unreliable server 77