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We consider the queueing system Mθ/G/1/∞ with two service modes. The server switches between the main service mode and threshold mode at the beginning of the current customer service when the number of customers present in the system is larger than h. The mean duration of the busy time and formulas for the stationary distribution of the number of customers in the system are obtained. 1. Description of the model The queueing system M/G/1/m with two service modes is studied in the paper [1]. The server switches between the modes at the beginning of the current customer service if the total number of customers in the system exceeds a given threshold level h. In the current paper, we use the results obtained in [1] to study a similar system without restrictions on the length of the queue, that is, we consider the case of m = ∞. Let there be given three sequences of independent identically distributed random variables {αn}, {θn}, and {βn}, n ≥ 1, where αn denotes the interarrival time between the (n − 1) and n batches of customers arriving to the system, θn denotes the number of customers in the n batch, and βn denotes the service time for the n th customer. Assume that P{αn < x} = 1− e−λx, λ > 0, and P{θn = i} = ai, i ≥ 1. If P{θn = 1} = a1 = 1, then each batch consists of only one customer. The customers are served one by one. After the service is completed, the customer leaves the system and the server starts the service of the next customer if there is at least one in the queue; otherwise, the server waits until a batch of customers arrives to the system. The FIFO discipline is used to maintain the queue in the system. The order of customers within a batch is arbitrary. Denote by ξ∞(t) the total number of customers in the system at time t. The phase space X∞ = {0, 1, . . . } corresponds to the stochastic process (ξ∞(t), t ≥ 0). For ξ∞(t), we introduce a threshold level h, h ≥ 1. Namely, if t is the moment when the service of the n customer starts and if ξ∞(t) ≤ h, then P{βn < x} = F (x) for x ≥ 0 (we agree that F (0) = 0). Otherwise, that is if ξ∞(t) > h, then P{βn < x} = F̃ (x), x ≥ 0, F̃ (0) = 0. 2010 Mathematics Subject Classification. Primary 60K25; Secondary 60K20.

Busy period and stationary distribution for the queueing system $\mathbf {M^{\theta }/G/1/\infty }$ with a threshold switching between service modes