A TWO-PRIORITY SINGLE SERVER RETRIAL QUEUE WITH ADDITIONAL ITEMS

. In this paper, we study a priority queueing-inventory problem with two types of customers. Arrival of customers follows Marked Markovian ar- rival process and service times have phase-type distribution with parameters depending on the type of customer in service. For service of each type of customer, a certain number of additional items are needed. High priority customers do not have waiting space and so leave the system when on their arrival a priority 1 customer is in service or the number of available additional items is less than the required threshold. Preemptive priority is assumed. Type 2 customers, encountering a busy server or idle with the number of available additional items less than a threshold, go to an orbit of inﬁnite capacity to retry for service. The customers in orbit are non-persistent: if on retrial the server is found to be busy/idle with the number of additional items less than the threshold, this customer abandons the system with certain probability. Such a system represents an accurate enough model of many real-world systems, including wireless sensor networks and system of cognitive radio with energy harvesting and healthcare systems. The probability distribution of the system states is computed, using which several of the characteristics are derived. A detailed numerical study of the system, including the analysis of the inﬂuence of the threshold, is performed.


(Communicated by Wuyi Yue)
Abstract. In this paper, we study a priority queueing-inventory problem with two types of customers. Arrival of customers follows Marked Markovian arrival process and service times have phase-type distribution with parameters depending on the type of customer in service. For service of each type of customer, a certain number of additional items are needed. High priority customers do not have waiting space and so leave the system when on their arrival a priority 1 customer is in service or the number of available additional items is less than the required threshold. Preemptive priority is assumed. Type 2 customers, encountering a busy server or idle with the number of available additional items less than a threshold, go to an orbit of infinite capacity to retry for service. The customers in orbit are non-persistent: if on retrial the server is found to be busy/idle with the number of additional items less than the threshold, this customer abandons the system with certain probability. Such a system represents an accurate enough model of many real-world systems, including wireless sensor networks and system of cognitive radio with energy harvesting and healthcare systems. The probability distribution of the system states is computed, using which several of the characteristics are derived. A detailed numerical study of the system, including the analysis of the influence of the threshold, is performed.
1. Introduction. In classical/retrial queue service is rendered to customers without regard to factors such as items required for providing service. This is equivalent to assuming an abundance of availability of such items at the service facility. However, if the availability of the additional items is not always guaranteed, then the customers will have to either wait or quit the system. This situation prevails in service facilities such as hospitals, automobile service stations, wholesale and retail shops and so on.
Motivated by the above-indicated real-life situations, in this paper we consider a retrial queue with a single server. High priority arriving customers get into service preempting the service of the low priority customer, provided the one in service is of low priority and the minimum required number of additional items for the service of the former is available. The preempted low priority customer is lost to the system. Additional items used for the service of a customer cannot be reused. They are discarded. These additional items are supplied to the system according to a Markovian arrival process (M AP ). The arrival of customers to the system is according to a marked Markovian arrival process (M M AP ). Low priority customers in the external arrival stream will join for service immediately provided the server is idle and a minimum level of additional items is available in stock; else such customers are directed to an orbit of infinite capacity from which they try to access the server with inter-retrial time exponentially distributed. Such retrial customers have a tendency to abandon the system with certain probability whenever the server is found to be busy/server idle with less than the minimum level of additional items. With complimentary probability, the customer gets back to the orbit. This process of abandonment of the system by low priority customers on unsuccessful retrial, coupled with linear retrial makes the system stable if the mentioned probability is greater than zero.
We give three real-life examples of the problem being investigated.
1. Wireless sensors with energy harvesting : Let us consider a sensor that monitors some area or the state of some equipment. Modern sensors have an opportunity to harvest energy from the outside (e.g. using solar, radiofrequency or wind energy) and store it. It is assumed that such stored energy is divided into energy units (that correspond to additional items in our model). Several energy units are required for transmission of one information unit. We assume that the sensor transmits to the central node two types of information units. Type 1 information unit (type 1 customer) has primary importance. Such information corresponds to the cases when someone invades into the secured area or the critically important equipment encounter a breakdown. Type 2 information unit (type 2 customer) corresponds to non-important routine information about the object parameters. If type 1 information arrives, the sensor immediately stops the transmission of routine information and starts the transmission of important information. Thus, type 1 customers have an absolute (preemptive) priority over type 2 customers. We assume that type 2 information that cannot be transmitted upon arrival may be regenerated (customers make retrials). To account the possibility of obsolescence of information we assume that type 2 customers are non-persistent.
The queueing system considered in our paper can help to optimally organize the operation of the sensor. For more information and the state-of-art in investigations of systems with energy harvesting see, e.g., [1,8,25]. 2. Cognitive radio systems with energy harvesting : In cognitive radio systems, primary users have the preemptive priority over the secondary users. Arrivals of the primary user cause immediate interruption to the service of secondary user. The secondary user, which encounters a busy server, retry for service. For more information about application of queueing theory to analysis of cognitive radio systems see, e.g., [11,28]. If this radio system is autonomous and harvests the energy for the surrounding environment, then the operation of the system is described by the model considered in this paper and the additional items are the energy units. A possibility of service of the secondary users increases the efficiency of the use of the equipment and profit of the service provider. However, it is necessary to elaborate on the optimal strategy of the secondary users admission, taking into account the amount of available harvested energy in particular. 3. Applications for modelling healthcare problems, in particular, planning and scheduling of operation theater : We model the work of an operation theater in a hospital. This room and the team of surgeons can be considered as the servers for patients service. There are two kinds of arriving patients (customers): a) emergency or non-elective patients for whom surgery is unexpected and hence needs to be performed urgently; b) elective patients, service of which can be planned well in advance and be temporarily postponed, see, e.g., [5]. The arrival of a non-elective patient implies a break of treatment of the elective patients, if it is currently provided. In the case if the service is provided to a non-elective patient, the new non-elective patient is rejected (he/she is urgently delivered to another hospital). As the additional items required for service provisioning, the following ones can be considered: intensive care unit (ICU) staff, ICU beds, stents, blood, etc. The number of the additional items required can be different for emergency and elective patients and there is a necessity to have some stock of the items dedicated for emergency cases. The elective customers whose ongoing service is canceled due to the emergency patient's arrival, treating can retry to arrange a new appointment later on.
The problem described above with orbit replaced by waiting room of infinite capacity for type 2 customers and abandonment of type 2 customers, is discussed in [3].
Retrial is a typical feature of many real-world systems, see, e.g. surveys [2,12,18,26]. Retrial queues with heterogeneous customers and priority are considered, e.g., in [4,7,9,13,14,23,27]. The most popular assumption in these papers is that the priority customers, which arrive when the server is busy, are queued in a finite or infinite buffer, while the non-priority customers in such a situation move to the orbit. But in [14] it is assumed that there is no buffer for high priority customers. The high priority customer, which arrives when the server is busy, interrupts the current service if the service is provided to a low priority customer, or rejected otherwise. With motivation stemming from application to the design of wireless sensor networks and healthcare system presented above we proceed with the assumptions indicated there. Possibility of the use of additional items for providing service is not assumed in the papers listed above.
For papers devoted to the analysis of the priority systems with the use of additional items for service of heterogeneous customers we refer to [24] and [29]. Effect of retrials is not considered in these papers. In [24], service time of a customer is assumed to be equal to zero.
For M AP and M M AP one may refer to the review article by Chakravarthy [6]. Kim et al. [16] and Kim et al. [17] interpret additional items required for service as a token needed to start service. Similar queueing models arise in the analysis of so-called queueing/inventory models, see [20], [21], [22]. This paper is arranged as follows. In Section 2, the model under study is described; also we investigate the stability of the system. For the stable system, we derive the system state distribution. Performance indices of the system are given in Section 3. Performance measures are numerically illustrated in Section 4. Finally, we conclude the paper indicating a plan of future extension of the model described.
2. Mathematical formulation. We consider a single-server queueing system with two types of customers, no buffer for type 1 customers, an infinite orbit for type 2 customers and a stock of capacity K for additional items. The structure of the system under study is given in Figure A.  [15] for its definition). The customers in the M M AP are heterogeneous and so of different types. The arrival of customers is directed by the stochastic process {ν(t), t ≥ 0} which is an irreducible continuous-time Markov chain with the state space {1, 2, ..., w}. The sojourn time of this chain in the state ν is exponentially distributed with parameter λ (ν) . When the sojourn time in the state ν expires, with probability p (0) ν,ν the process {ν(t), t ≥ 0} jumps to the state ν without generation of a customer, ν, ν ∈ {1, 2, ..., w}, ν = ν and with probability p ( ) ν,ν the process {ν(t), t ≥ 0} jumps to the state ν with the generation of type customer, = 1, 2; ν, ν ∈ {1, 2, ..., w}. The behaviour of the M M AP is completely characterized by the matrices D , = 0, 1, 2, defined by = 1, 2. The matrix D = D 0 + D 1 + D 2 represents the generator of the process {ν(t), t ≥ 0}.
The average total arrival intensity λ is defined by λ = θ(D 1 + D 2 )e where θ is the invariant vector of the stationary distribution of the Markov chain {ν(t), t ≥ 0}. The vector θ is the unique solution to the system θD = 0 and θe = 1. The average arrival intensity λ of type customers is defined by λ = θD e, = 1, 2. Here, e is a column vector of 1 s of appropriate order.
The squared overall (without differentiating the types of customers) coefficient of variation of intervals between successive arrivals is given as The squared coefficient of variation of inter-arrival times of type customers is given as The coefficient of correlation of two successive intervals between arrivals is given as We assume that K additional items are required for service of each type customer, = 1, 2. So, at the moment when type customer is chosen for service, the number of additional items in the stock decreases by K , = 1, 2.
Type 1 customers are assumed to be priority customers and have limited preemptive priority over type 2 customers. Arriving type 1 customer is not accepted for service (is lost) only if the server provides service to another type 1 customer or the number of additional items in the stock is less than K 1 . If the server provides service to type 2 customer at the moment of arrival of type 1 customer, service of type 2 customer is immediately terminated, it leaves the system permanently (is lost) and the engaged into service additional items are lost as well provided there are at least K 1 additional items available.
Different scenarios of the system behaviour during an arbitrary type 1 customer arrival epoch are: 1. the server is busy with type 1 customer, the arriving type 1 customer is lost. 2. the number of additional items in the stock is less than K 1 , the arriving type 1 customer is lost. If at this moment the server is providing service to type 2 customer, the service is not terminated. 3. the number of additional items in the stock is greater than or equal to K 1 and the server is free, the arriving type 1 customer occupies the server. 4. the number of additional items in the stock is greater than or equal to K 1 and the server is busy with type 2 customer, the arriving type 1 customer occupies the server. Type 2 customer, whose service was terminated, is lost. Arriving type 2 customer can start service only if the server is idle and the number of additional items in the stock is greater than or equal to some preassigned threshold N, N ≥ K 2 . Otherwise, this customer joins the orbit of infinite capacity from where he makes repeated attempts for service according to an exponentially distributed time with parameter nγ when n customers are in the orbit. On successful retrial (server found idle and the number of available additional items is not less than N ) he joins for service. If the retrial is a failure then with probability q, he leaves the system forever, and with complimentary probability he returns to orbit. The threshold N is an important control parameter. If N = K 2 , then the arriving type 2 customer starts service when the server is idle. If N > K 2 , then the type 2 customer cannot start service even when the server is idle and there are additional items for its service but the number of additional items is less than N. The idea of introducing this control parameter N is to have some reserve of items for the case when type 1 customer arrives. The choice of N is not trivial. If N is too large, the server is not effectively used. Frequently it does not work even when there is a lot of type 2 customers in the orbit. Also, a large value of N implies the higher probability that the stock for additional items will be overloaded and arriving items will be lost. If N is too small (we make insufficient reservation of additional items in the stock), then the probability that the priority type 1 customer cannot start service due to the lack of additional items in the stock is large. One of the goals of the analysis presented in this paper is to highlight the influence of the threshold N.
Different scenarios of the system behaviour at an arbitrary type 2 customer arrival epoch are: 1. the number of additional items in the stock is greater than or equal to N, K 2 − 1 < N ≤ K, and the server is free, the arriving customer occupies the server and the number of additional items decreases by K 2 . 2. the number of additional items in the stock is less than N or the server is busy, the arriving type 2 customer proceeds to the orbit.
Additional items arrive at the system according to the Markovian arrival process (MAP). Arrival in the MAP (see Chakravarthy [6]) is a special class of semi-Markov process with underlying continuous-time Markov chain (CT M C), {ζ(t), t ≥ 0}, on the state space {1, 2, ..., m} with generator H = H 0 +H 1 such that H 0 governs transitions corresponding to no arrivals and H 1 accounts for transitions corresponding to arrivals. These matrices, H 0 and H 1 , are of the form given by ij , 1 ≤ i, j ≤ m, represents the rate of transition from i to j through an arrival, while h (0) ij , 1 ≤ i, j ≤ m, i = j, represents the rate of transition from i to j without an arrival. Note that transition from i to i is possible only through an arrival and not otherwise. Let η be the steadystate probability vector of H. Then ηH = 0 and ηe = 1. The fundamental rate λ R of this MAP is given by λ R = ηH 1 e which gives the expected number of arrivals per unit of time. The coefficient of variation c var of intervals between arrivals is calculated as c var = 2λ R η(−H 0 ) −1 e − 1 and the coefficient of correlation c cor of intervals between successive arrivals is given as If at the arrival epoch of additional item the stock of items is full, then the arriving item is lost.
The service time of an arbitrary type customer has phase-type distribution with an irreducible representation (β , S ), = 1, 2.
The squared coefficient of variation is given by Under the above assumption we consider continuous-time Markov chain is the number of type 2 customers in the orbit; S(t) is the status of server : 1 if server busy with type 1 customer, 2 if server busy with type 2 customer; I(t) is the number of additional items; ν(t) is the state of the underlying MMAP; ζ(t) is the state of the underlying MAP; η(t) is state of the phase-type service time at the moment t, t ≥ 0.
The Markov chain {ξ(t), t ≥ 0} has the following state space In the sequel we need the following notations and abbreviations: • e j = Column vector of 1 s of order j.
• I = Identity matrix of appropriate order.
• I j = Identity matrix of order j.
Lemma 2.1. The infinitesimal generator Q of the Markov chain {ξ(t), t ≥ 0} describing the model, has a block-tridiagonal structure which is of the form where each sub-matrix is of order wm(K + 1)(1 + M 1 + M 2 ).
n,n−1 = I K+1 ⊗ nqγI wmM for n ≥ 1, = 1, 2, Proof. The proof is implemented by the analysis of all transitions of the Markov chain {ξ(t), t ≥ 0} during an infinitesimal interval. Then we rewrite the intensities of these transitions in block matrix form. The infinitesimal generator Q has a block-tridiagonal structure because no more than one arrival can occur and no more than one service can be completed in the system during the small interval of negligible size. That is, The entries of the blocks Q n,n−1 for n ≥ 1, define the intensities of the events that decrease the number of type 2 customers in the orbit by one.
The non-diagonal entries of the blocks Q n,n for n ≥ 0, define the intensities of the events of the Markov chain {ξ(t), t ≥ 0} that do not change the number of type 2 customers in the orbit.
The entries of the matrix Q 0 define the intensities of the transitions that increase the number of type 2 customers in the orbit by one.
The Markov chain {ξ(t), t ≥ 0} belongs to the class of continuous-time AQTMC. From [19], the sufficient condition for the ergodicity of AQT M C {ξ(t), t ≥ 0} is expressed in terms of the matrices as follows: where R n is a diagonal matrix with diagonal entries defined as the moduli of the corresponding diagonal entries of the matrix Q n,n , n ≥ 0.
Here, R n is the block diagonal matrix with the diagonal blocks R ( ) n , n ≥ 0, = 0, 1, 2, defined as follows: where the matricesF l , l = 0, 1, 2, are the diagonal matrices the diagonal entries of which coincide with the diagonal entries of the matrices F l , l = 0, 1, 2, correspondingly:F The explicit forms of the square matrices Y , = 0, 1, 2 are as follows: • If q = 0, then otherwise.
• If q = 0, then 2900 DHANYA SHAJIN, A. N. DUDIN, OLGA DUDINA AND A. KRISHNAMOORTHY As proved in [19], the stationary state distribution of the Markov chain {ξ(t), t ≥ 0} exists if the inequality yY 2 e < yY 0 e (2) holds true, where the row vector y is the unique solution to the following system of linear algebraic equations If q = 0, then the condition (2) can be rewritten as and it always holds true if the vector y satisfies system (3) rewritten as yY 0 = y, ye = 1.
If q = 0, by direct substitution into (3), it can be verified that the solution to system (3) has the following form: where y satisfy the following relations: Hence we have From the normalizing condition Thus we have proved the following theorem Theorem 2.2. If q = 0, then the Markov chain {ξ(t), t ≥ 0} is ergodic for any choice of system parameters.
It is well known that the probability vectors x n , n ≥ 0, satisfy the following system of linear equations: xQ = 0 and xe = 1, where Q is the infinitesimal generator of {ξ(t), t ≥ 0}. Taking into account that the fact the matrix Q has a block-tridiagonal structure, to solve the system of equations given in (9), we propose the following numerically stable algorithm (see [19]). Theorem 2.3. The vectors x n , n ≥ 0 are calculated as where the matrices F n are calculated using the recurrent formulas the matrices G n are computed using the backward recursion and the vector x 0 is the unique solution to the system As n → ∞ the sequence of matrices G n tends to the matrix G which is the minimal non-negative solution to the matrix equation (see Klimenok and Dudin [19]) Next, we enumerate a few characteristics of the system that throw sufficient light on the performance of the system.

Performance measures.
1. The probability that at an arbitrary epoch the server is busy with type 1 customers: x(n, 1)e.
2. The probability that at an arbitrary epoch the server is busy with type 2 customers: x(n, 2)e.
3. The average number of customers in the orbit: nx n e.

The average number of additional items in the stock:
5. The average intensity of flow of type customers who receive service: 6. The probability that an arbitrary type customer will be lost: 7. The probability of loss of an arbitrary type 1 customer because the server is busy with type 1 customer: 8. The probability of loss of an arbitrary arriving type 1 customer due to lack of additional items: 9. The probability that service of an arbitrary type 2 customer is terminated by an arriving type 1 customer: 10. The probability that an arbitrary additional item will be lost: 11. The probability of an arbitrary type 1 arrival occupies the server upon arrival: 12. The probability of an arbitrary arriving type 2 customer occupies the server upon arrival: 13. The probability that the server is idle: x(n, 0)e.
14. The probability that the server is idle with customers in the orbit: 15. Effective rate at which the customers leave the system after unsuccessful retrial: 16. Effective rate at which the customers enter into service after successful retrial: Next we provide a few computational results that provide indepth understanding of the system.

Numerical illustration.
Numerical experiments have essentially 4 goals. These are to illustrate: • Effect of variation of N.
• Effect of variation of q.
• Effect of correlation in the arrival process of customers and additional items.
• Effect of retrial rate γ and loss probability q. We assume that the stock of the additional items has capacity K = 20, the number of additional items required for service of type 1 customer is K 1 = 5 and the number of additional items required for service of type 2 customer is K 2 = 2. The retrial rate γ = 2. PH service process of type 1 customers are characterized by In this case the mean service time b  Table 1 and Figure 1 provide dependence of the average number N O of type 2 customers in the orbit and the average number N item of additional items in the system for distinct combinations of arrival processes of additional items and that of customers for various values of the minimum number N of required additional items. These two characteristics increase with the value of N. This is easily explained. The growth of N implies the decrease of the chances of type 2 customer's entering the service upon arrival. This leads to the increase in the average number of type 2 customers in the orbit, and, hence, the probability of these customers loss due to non-persistency. Thus, fewer items are used for the service of type 2 customers. In turn, this implies the increase in the average number of additional items in the system. Note, that the difference between the highest and lowest values of N item is maximum for the combination M AP 0.4 + M M AP 0 . Figure 2 along with Table 2 provide the graphical and numerical illustration of behaviour of type 1 customer loss probability, p loss 1 , and that of an arriving additional item being lost, p loss item . The growth of N evidently creates better conditions for type 1 customers that results in an increase of p loss 1 . The increase of p loss item stems from the increase of the average number of items in the stock that was explained above. Table 3 and Figure 3 show that the probability p lack loss 1 of type 1 customer loss due to the lack of items significantly decreases with grows of N. This agrees with observations made above that the growth of N implies larger average number of items in the stock. The probability p busy loss 1 of type 1 customer loss, because this customer arrives when the service is provided to another type 1 customer, increases with the growth of N. This stems from the decrease of the probability of type 1 customer loss due to lack of items and higher probability that type 1 customer is in service at an arbitrary moment. In all descriptions above q was fixed at 0.2.   It is evidently seen from Figures 1-3 that correlation in arrival processes of customers and additional items has a very significant effect on the values of performance measures of the system. Next, we vary over q and examine the dependence of N O and N item on it. With q increasing (see Table 4 and Figure 4) all combinations M AP + M M AP result in reduction in type 2 customers in the orbit, which is highly significant for M AP 0.4 + M M AP 0.4 combination. The reneging probability q thus plays a very crucial role in determining the number of orbital customers in the system. It is clear that the increase of q leads to more intensive loss of type 2 customers from the orbit and therefore, to the essential decrease of the average number of customers in orbit, and the increase of the number of items in the stock.
Dependence of the loss probability p loss 1 of type 1 customer and the loss probability p loss item of an arbitrary additional item are provided in Table 5 and Figure 5.    Figure 2. Dependence of probability that an arbitrary type 1 customer will be lost p loss 1 and probability that an arbitrary additional item will be lost p loss item on N The loss probability of additional item increases with increase in the value of q (see Table 5b and Figure 5b). This is because additional items are not intensively used by type 2 customers (with increase of q the loss probability of type 2 customers evidently grows). The growth of the number of additional items implies the decrease of the loss probability of type 1 customers. Table 6 and Figure 6 provide the dependence on q of loss probability of a type 1 customer due to server being busy with a customer of the same type and that due to unavailability of required number of additional items, with increasing value of q. It may be noted that p busy loss 1 increases for all combination of M AP &M M AP (see Table 6a) with increase in value of q. Availability of additional items increases, thus, p lack loss 1 turns out to be decreasing (see Table 6b). In Table 7 and Figure 7 the combined effect of retrial rate and abandoning probability on the number of customers (type 2) in the orbit is illustrated. We   Figure 3. Dependence of probability of an arbitrary arriving type 1 customer loss because the server is busy with type 1 customer p busy loss 1 and probability of an arbitrary arriving type 1 customer loss due to lack of additional items p lack loss 1 on N adjusted q and γ values such that the product of γ and q in each row is the same. With q increasing and γ decreasing such that product qγ = 0.15, we see that N O decreases for all arrival combinations M AP and M M AP of additional items and customers.
Conclusion. We analyzed a priority queueing-inventory problem with two types of customers. Arrivals of customers follow marked Markovian arrival process and service time durations are phase-type distributed with representation (β 1 , S 1 ) of order M 1 and (β 2 , S 2 ) of order M 2 respectively for type 1 and type 2 customers, which are mutually independent. For service of each type of customer, a certain minimum number of additional items are needed. High priority customers do not have waiting space and so leave the system when on its arrival a priority 1 is    found in service/number of additional items available for its service is less than the required (when type 2 is in service). Preemption priority is assumed. Type 2 customers, encountering a busy server, go to an orbit of infinite capacity to retry for service; even when the server is idle but the number of additional items is less than the threshold, then again low priority customers go to orbit. If on retrial, the server is found to be busy/idle with the number of additional items less than the threshold, the low priority customer abandons the system with probability q; with complementary probability, it gets back to orbit. The ergodicity condition is derived. The system state probability is computed from which several characteristics of the system are derived. A detailed numerical analysis of the system is performed. Extending the model discussed to the multi-server case is proposed as future work. This will be quite challenging.    Figure 5. Dependence of probability that an arbitrary type 1 customer will be lost p loss 1 and probability that an arbitrary additional item will be lost p loss item on q    Figure 6. Dependence of probability of an arbitrary arriving type 1 customer loss because the server is busy with type 1 customer p busy loss 1 and probability of an arbitrary arriving type 1 customer loss due to lack of additional items p lack loss 1 on q